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A LINNTHIT- PRIVATE SCHOOL A LINNTHIT- PRIVATE SCHOOL ……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
Koaunglwinoo ( thufonesarr-education website ) Page | 1
B
H
M
EXERCISE 7.1
1.Let Maung Ba = B , Maung Hla = H , Maung Mya = M ,
Ma Ni = N , Ma Yi = Y , Ma Thi = T , Ma Si = S
President Vice President Possible Outcomes
N ( B, N )
Y ( B, Y )
T ( B, T )
S ( B, S )
N ( H, N )
Y ( H, Y )
T ( H, T )
S ( H, S )
N ( M, N )
Y ( M, Y )
T ( M, T )
S ( M, S )
P ( Maung Ba is to be selected for presidents ) = ?
The set of all possible outcomes = { ( B , N ) , ( B, Y ) , ( B, T ) ,( B, S ) ,( H, N ) ,( H,
Y ), ( H, T )
( H, S ) , ( M, N ) , ( M, Y ) , ( M, T ) , (
M, S ) }
Number of possible outcomes = 12
The set of favourable outcomes = { ( B , N ) , ( B, Y ) , ( B, T ) ,( B, S ) }
Number of favourable outcomes = 4
P ( A ) = 𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐟𝐚𝐯𝐨𝐮𝐫𝐚𝐛𝐥𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬
𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐩𝐨𝐬𝐬𝐢𝐛𝐥𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬
P ( Maung Ba is to be selected for presidents ) = 𝟒
𝟏𝟐 =
𝟏
𝟑
---------------------------------------------------------------------
2. replaced ပစာၦ
1st Choice 2nd Choice Possible Outcomes
2 ( 2 , 2 )
3 ( 2 , 3 )
2 4 ( 2 , 4 )
5 ( 2 , 5 )
9 ( 2 , 9 )
2 ( 3 , 2 )
3 ( 3 , 3 )
3 4 ( 3 , 4 )
5 ( 3 , 5 )
9 ( 3 , 9 )
2 ( 4 , 2 )
3 ( 4 , 3 )
4 4 ( 4 , 4 )
5 ( 4 , 5 )
9 ( 4 , 9 )
2 ( 5 , 2 )
3 ( 5 , 3 )
5 4 ( 5 , 4 )
5 ( 5 , 5 )
9 ( 5 , 9 )
2 ( 9 , 2 )
3 ( 9 , 3 )
9 4 ( 9 , 4 )
5 ( 9 , 5 )
9 ( 9 , 9 )
The set of all possible outcomes ={ ( 2 , 2 ),( 2 , 3 ),( 2 , 4 ) ,( 2 , 5 ), ( 2 , 9 )
,( 3 , 2 ),( 3 , 3 ) ,( 3 , 4 ),( 3 , 5 ),( 3 , 9 ) ,( 4 , 2 ) ,( 4 , 3 ) ,( 4 , 4 ) ,( 4 , 5 )
,( 4 , 9 ),( 5 , 2 ) ,( 5 , 3 ) ,( 5 , 4 ) ,( 5 , 5 ) ,( 5 , 9 ) , ( 9 , 2 ),( 9 , 3 ),( 9 , 4 )
,( 9 , 5 ),( 9 , 9 )}
Number of possible outcomes = 25
( a ) P ( getting two prime numbers ) = ?
A LINNTHIT- PRIVATE SCHOOL A LINNTHIT- PRIVATE SCHOOL ……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
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**( ေရ႕ကနးေရာ ေနာကကနးေရာ prime number မား ျဖစေနမည ျဖစတနစြမး )
The set of favourable outcomes ={ ( 2 , 2 ),( 2 , 3 ),( 2 , 5 ),( 3 , 2 ),( 3 , 3 )
,( 3 , 5 ) ,( 5 , 2 ) ,( 5 , 3 ),( 5 , 5 ) }
Number of favourable outcomes = 9
P ( A ) = 𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐟𝐚𝐯𝐨𝐮𝐫𝐚𝐛𝐥𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬
𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐩𝐨𝐬𝐬𝐢𝐛𝐥𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬
P ( getting two prime numbers ) = 𝟗
𝟐𝟓
( b ) P ( getting two odd numbers ) = ? ( ေရ႕ကနး ေနာကကနးမကနးမား ျဖစေနမည ျဖစတနစြမး )
The set of favourable outcomes ={ ( 3 , 3 ),( 3 , 5 ),( 3 , 9 ),( 5 , 3 ),( 5 , 5 )
,( 5 , 9 ) ,( 9 , 3 ),( 9 , 5 ),( 9 , 9 ) }
Number of favourable outcomes = 9
P ( getting two prime numbers ) = 𝟗
𝟐𝟓
( c ) P ( getting a pair of numbers where the sum is a prime numbers ) = ?
**( ေရ႕ကနးႏင ေနာကကနးေပါငးျခငးတနဖး prime number ျဖစေနမည ျဖစတနစြမး )
The set of favourable outcomes ={ ( 2 , 3 ),( 2 , 5 ),( 2 , 9 ) ,( 3 , 2 ),( 3 , 4 )
,( 4 , 3 ),( 4 , 9 ),( 5 , 2 ),( 9 , 2 ),( 9 , 4 ) }
Number of favourable outcomes = 10
P ( getting a pair of numbers where the sum is a prime numbers ) = 𝟏𝟎
𝟐𝟓
---------------------------------------------------------------------
3. not replaced ပစာၦ
1st Choice 2nd Choice Possible Outcomes
3 ( 2 , 3 )
4 ( 2 , 4 )
5 ( 2 , 5 )
9 ( 2 , 9 )
2 ( 3 , 2 )
4 ( 3 , 4 )
5 ( 3 , 5 )
9 ( 3 , 9 )
2 ( 4 , 2 )
3 ( 4 , 3 )
5 ( 4 , 5 )
9 ( 4 , 9 )
2 ( 5 , 2 )
3 ( 5 , 3 )
4 ( 5 , 4 )
9 ( 5 , 9 )
2 ( 9 , 2 )
3 ( 9 , 3 )
4 ( 9 , 4 )
5 ( 9 , 5 )
The set of all possible outcomes ={ ( ( 2 , 3 ),( 2 , 4 ),( 2 , 5 ),( 2 , 9 )
,( 3 , 2 ),( 3 , 4 ),( 3 , 5 ),( 3 , 9 ) ,( 4 , 2 ) ,( 4 , 3 ),( 4 , 5 ),( 4 , 9 )
,( 5 , 2 ) ,( 5 , 3 ) ,( 5 , 4 ),( 5 , 9 ), ( 9 , 2 ),( 9 , 3 ),( 9 , 4 ),( 9 , 5 )}
Number of possible outcomes = 25
( a ) P ( getting two prime numbers ) = ?
The set of favourable outcomes ={ ( 2 , 3 ),( 2 , 5 ),( 3 , 2 )
, ( 3 , 5 ) ( 5 , 2 ) ,( 5 , 3 ) }
Number of favourable outcomes = 6
P ( A ) = 𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐟𝐚𝐯𝐨𝐮𝐫𝐚𝐛𝐥𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬
𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐩𝐨𝐬𝐬𝐢𝐛𝐥𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬
P ( getting two prime numbers ) = 𝟔
𝟐𝟓
( b ) P ( getting two odd numbers ) = ?
The set of favourable outcomes ={ ( 3 , 5 ),( 3 , 9 ),( 5 , 3 )
,( 5 , 9 ) ,( 9 , 3 ),( 9 , 5 ) }
Number of favourable outcomes = 6
P ( getting two prime numbers ) = 𝟔
𝟐𝟓
( c ) P ( getting a pair of numbers where the sum is a prime numbers ) = ?
The set of favourable outcomes ={ ( 2 , 3 ),( 2 , 5 ),( 2 , 9 ) ,( 3 , 2 ),( 3 , 4 )
2
3
4
5
9
A LINNTHIT- PRIVATE SCHOOL A LINNTHIT- PRIVATE SCHOOL ……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
Koaunglwinoo ( thufonesarr-education website ) Page | 3
,( 4 , 3 ),( 4 , 9 ),( 5 , 2 ),( 9 , 2 ),( 9 , 4 ) }
Number of favourable outcomes = 10
P ( getting a pair of numbers where the sum is a prime numbers ) = 𝟏𝟎
𝟐𝟓
---------------------------------------------------------------------
4. not replaced ပစာၦ
Let the 2 blue marble = b1 , b
2 , 1 red marble = r , 1 yellwo marble = y
1st Choice 2nd Choice Possible Outcomes
b2 ( b
1 , b
2 )
b1 r ( b
1 , r )
y ( b1 , y )
b1 ( b
2 , b
1 )
b2 r ( b
2 , r )
y ( b2 , y )
b1 ( r , b
1 )
r b2 ( r , b
2 )
y ( r , y )
b1 ( r , b
1 )
y b2 ( r , b
2 )
r ( y , r )
The set of all possible outcomes = {( b1 , b
2 ),( b
1 , r ),( b
1 , y ),( b
2 , b
1 )
,( b2 , r ), ( b
2 , y ),( r , b
1 ),( r , b
2 ),( r , y ),( r , b
1 ),( r , b
2 ),( y , r ) }
Number of possible outcomes = 12
( a ) P ( choosing 2 blue marbles ) = ?
The set of favourable outcomes = { ( b1 , b
2 ),( b
2 , b
1 ) }
Number of favourable outcomes = 2
P ( A ) = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
P ( choosing 2 blue marbles ) = 𝟐
𝟏𝟐 =
𝟏
𝟔
( b )P ( choosing 2 different colours ) = ?
The set of favourable outcomes = { ( b1 , r ),( b
1 , y ),( b
2 , r ), ( b
2 , y )
,( r , b1 ),( r , b
2 ) ,( r , y ),( r , b
1 ),( r , b
2 ),( y , r ) }
Number of favourable outcomes = 10
P ( choosing 2 different colours ) = 𝟏𝟎
𝟏𝟐 =
𝟓
𝟔
---------------------------------------------------------------------
5. 1st Spin 2nd Spin Possible Outcomes
B ( B , B )
B R ( B , R )
Y ( B , Y )
B ( R , B )
R R ( R , R )
Y ( R , Y )
B ( Y , B )
Y R ( Y , R )
Y ( Y , Y )
ျမားလညသည ပစာၦတငး
သည replaced ပစာၦ
မားႏင သေဘာအတ
တပငျဖစသည။
A LINNTHIT- PRIVATE SCHOOL A LINNTHIT- PRIVATE SCHOOL ……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
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The set of all possible outcomes = { ( B , B ),( B , R ),( B , Y ),( R , B ),( R , R )
,( R , Y ),( Y , Y ),( Y , R ),( Y , B ) }
Number of possible outcomes = 9
( a ) P ( not spinning red first ) = ? ( ေရ႕အေရာင အနမျဖစသည ျဖစတနစြမး )
The set of favourable outcomes ={( B , B ),( B , R ),( B , Y ),( Y , Y ),( Y , R ),( Y , B ) }
Number of favourable outcomes = 6
P ( A ) = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
P ( not spinning red first ) = 𝟔
𝟗 =
𝟐
𝟑
( a ) P ( spinning 2 different colours ) = ?
The set of favourable outcomes ={( B , R ),( B , Y ),( R , B ),( R , Y ),( Y , B ),( Y , R ) }
Number of favourable outcomes = 6
P (spinning 2 different colours ) = 𝟔
𝟗 =
𝟐
𝟑
---------------------------------------------------------------------
6. ဆေပးျခငး ဥကဌ ေရြးျခငး ျဖစရပမားသည not replaced ျဖစရပမား ျဖစၾကပါသညၤ။
Let Maung Maung = Maung , Maung Mya = Mya , Ma Hla = Hla , Ma Khin = Khin
1st Prize 2nd Prize Possible Outcomes
Mya ( Maung , Mya )
Maung Hla ( Maung , Hla )
Khin ( Maung , Khin )
Maung ( Mya , Maung )
Mya Hla ( Mya , Hla )
Khin ( Mya , Khin )
Maung ( Hla , Maung )
Hla Mya ( Hla , Mya )
Khin ( Hla , Khin )
Maung ( Khin , Maung )
Khin Mya ( Khin , Mya )
Hla ( Khin , Hla )
The set of all possible outcomes ={ ( Maung , Mya ),( Maung , Hla ), ( Maung , Khin )
,( Mya , Maung ) ,( Mya , Hla ),( Mya , Khin )
,( Hla , Maung ),( Hla , Mya ), ( Hla , Khin )
,( Khin , Maung ), ( Khin , Mya ), ( Khin , Hla ) }
Number of possible outcomes = 12
P ( Maung Mya and Ma Khin both win prizes ) = ?
P ( A ) = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
The set of favourable outcomes = { ( Mya , Khin ) , ( Khin , Mya ) }
Number of favourable outcomes = 2
P ( Maung Mya and Ma Khin both win prizes ) = 2
12 =
1
6
---------------------------------------------------------------------
7. ဒဂၤြးျပားေျမာကျခငးျဖစရပမားသည replaced ျဖစရပမား ျဖစၾကပါသညၤ။
1st Toss 2nd Toss 2nd Toss Possible Outcomes
H ( H , H , H )
T ( H , H , T )
H ( H , T , H )
T ( H , T , T )
H ( T , H , H )
T ( T , H , T )
H ( T , T , H )
T ( T , T , T )
H
T
H
T
H
T
A LINNTHIT- PRIVATE SCHOOL A LINNTHIT- PRIVATE SCHOOL ……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
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The set of all possible outcomes = { ( H , H , H ),( H , H , T ),( H , T , H )
,( H , T , T ),( T , H , H ),( T , H , T ),( T , T , H ), ( T , T , T ) }
Number of possible outcomes = 8
( a ) P ( getting exactly one head ) = ?
( သးၾကမးေျမာကတာမာ ေခါငးတစၾကမတညး တတကကပါတ ျဖစတနစြမး )
The set of favourable outcomes = { ( H , T , T ), ( T , H , T ), ( T , T , H ) }
Number of favourable outcomes = 3
P ( A ) = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚 𝑒𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
P ( getting exactly one head ) = 3
8
( b ) P ( getting no head ) = ?
( သးၾကမးေျမာကတာမာ ေခါငးတစၾကမမ မကတ ျဖစတနစြမး )
The set of favourable outcomes = { ( T , T , T ) }
Number of favourable outcomes = 1
P ( getting no head ) = 1
8
---------------------------------------------------------------------
8.
Let boy = B , girl = G
1st Child 2nd Child 2nd Child Possible Outcomes
B ( B , B , B )
G ( B , B , G )
B ( B , G , B )
G ( B , G , G )
B ( G , B , B )
G ( G , B , G )
B ( G, G , B )
G ( G , G , G )
The set of all possible outcomes = { ( B , B , B ),( B , B , G ),( B , G , B )
,( B , G , G ),( G , B , B ),( G , B , G ),( G , G , B ),( G , G , G ) }
Number of possible outcomes = 8
( a ) P ( the first two children are boys ) = ?
The set of favourable outcomes = { ( B , B , B ),( B , B , G ) }
Number of favourable outcomes = 2
P ( A ) = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏 𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
P (the first two children are boys ) = 2
8 =
1
4
( b ) P (the first two children are boys ) = ?
( သးၾကမးေျမာကတာမာ ေခါငးတစၾကမမ မကတ ျဖစတနစြမး )
The set of favourable outcomes = { ( B , B , B ),( G , B , B ) }
Number of favourable outcomes = 2
P ( getting no head ) = 2
8 =
1
4
---------------------------------------------------------------------
B
G
B
G
B
G
သးေယာကေမြးတာမာ
ပထမႏစေယာက
ေယာကာေလး ျဖစတနစြမး
A LINNTHIT- PRIVATE SCHOOL A LINNTHIT- PRIVATE SCHOOL ……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
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8.
Tossing Co[n 2nd
Thrown 2nd
Thrown Possible Outcomes Possible Outcomes 1 ( H , 1 , 1 ) ( T , 1 , 1 ) 2 ( H , 1 , 2 ) ( T , 1 , 2 ) 3 ( H , 1 , 3 ) ( T , 1 , 3 ) 4 ( H , 1 , 4 ) ( T , 1 , 4 ) 5 ( H , 1 , 5 ) ( T , 1 , 5 ) 6 ( H , 1 , 6 ) ( T , 1 , 6 ) 1 ( H , 2 , 1 ) ( T , 2 , 1 ) 2 ( H , 2 , 2 ) ( T , 2 , 2 ) 3 ( H , 2 , 3 ) ( T , 2 , 3 ) 4 ( H , 2 , 4 ) ( T , 2 , 4 ) 5 ( H , 2 , 5 ) ( T , 2 , 5 ) 6 ( H , 2 , 6 ) ( T , 2 , 6 ) 1 ( H , 3 , 1 ) ( T , 3 , 1 ) 2 ( H , 3 , 2 ) ( T , 3 , 2 ) 3 ( H , 3 , 3 ) ( T , 3 , 3 ) 4 ( H , 3 , 4 ) ( T , 3 , 4 ) 5 ( H , 3 , 5 ) ( T , 3 , 5 ) 6 ( H , 3 , 6 ) ( T , 3 , 6 ) 1 ( H , 4 , 1 ) ( T , 4 , 1 ) 2 ( H , 4 , 2 ) ( T , 4 , 2 ) 3 ( H , 4 , 3 ) ( T , 4 , 3 ) 4 ( H , 4 , 4 ) ( T , 4 , 4 ) 5 ( H , 4 , 5 ) ( T , 4 , 5 ) 6 ( H , 4 , 6 ) ( T , 4 , 6 ) 1 ( H , 5 , 1 ) ( T , 5 , 1 ) 2 ( H , 5 , 2 ) ( T , 5 , 2 ) 3 ( H , 5 , 3 ) ( T , 5 , 3 ) 4 ( H , 5 , 4 ) ( T , 5 , 4 ) 5 ( H , 5 , 5 ) ( T , 5 , 5 ) 6 ( H , 5 , 6 ) ( T , 5 , 6 ) 1 ( H , 6 , 1 ) ( T , 6 , 1 ) 2 ( H , 6 , 2 ) ( T , 6 , 2 ) 3 ( H , 6 , 3 ) ( T , 6 , 3 ) 4 ( H , 6 , 4 ) ( T , 6 , 4 ) 5 ( H , 6 , 5 ) ( T , 6 , 5 ) 6 ( H , 6 , 6 ) ( T , 6 , 6 )
The set of all possible outcomes = { ( H , 1 , 1 ),( H , 1 , 2 ),( H , 1 , 3 ),( H , 1 , 4 )
,( H , 1 , 5 ),( H , 1 , 6 ),( H , 2 , 1 ),( H , 2 , 6 ),( H , 2 , 3 ),( H , 2 , 4 ),( H , 2 , 5 ),( H , 2 , 2 ),( H , 3 , 1 )
.( H , 3 , 2 ).( H , 3 , 3 ).( H , 3 , 4 ),( H , 3 , 5 ),( H , 3 , 6 ),( H , 4 , 1 ),( H , 4 , 2 ),( H , 4 , 3 ),( H , 4 , 4 )
,( H , 4 , 5 ),( H , 4 , 6 ),( H , 5 , 1 ),( H , 5 , 2 ),( H , 5 , 3 ),( H , 5 , 4 ),( H , 5 , 5 ),( H , 5 , 6 ),( H , 6 , 1 )
,( H , 6 , 2 ),( H , 6 , 3 ),( H , 6 , 6 ),( H , 6 , 5 ),( H , 6 , 4 ),( T , 1 , 1 ),( T , 1 , 2 ),( T , 1 , 3 ),( T , 1 , 4 )
,( T , 1 , 6 ),( T , 1 , 5 ),( T , 2 , 1 ),( T , 2 , 2 ),( T , 2 , 3 ),( T , 2 , 4 ),( T , 2 , 5 ),( T , 2 , 6 ),( T , 3 , 1 )
,( T , 3 , 2 ),( T , 3 , 3 ),( T , 3 , 4 ),( T , 3 , 5 ),( T , 3 , 6 ),( T , 4 , 1 ),( T , 4 , 2 ),( T , 4 , 3 ),( T , 4 , 4 )
,( T , 4 , 5 ),( T , 4 , 6 ),( T , 5 , 1 ),( T , 5 , 2 ),( T , 5 , 3 ),( T , 5 , 4 ),( T , 5 , 5 ),( T , 5 , 6 ),( T , 6 , 1 )
,( T , 6 , 2 ),( T , 6 , 6 ),( T , 6 , 4 ),( T , 6 , 5 ),( T , 6 , 3 ) }
Number of possible outcomes = 72
P ( Head and 6 turn up ) = ?
The set of favourable outcomes = { ( H , 1 , 6 ),( H , 2 , 6 ),( H , 3 , 6 )
,( H , 4 , 6 ),( H , 5 , 6 ), ( H , 6 , 1 )
,( H , 6 , 2 ),( H , 6 , 3 ),( H , 6 , 4 )
,( H , 6 , 5 ), ( H , 6 , 6 ) }
Number of favourable outcomes = 11
P ( A ) = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
P (Head and 6 turn up ) = 11
72
---------------------------------------------------------------------
Example ( 1 ) Black dies ( or ) 2nd die
The set of all possible outcome = { ( 1 , 1 ), ( 1 , 2 ), ( 1 , 3 ),( 1 , 4 ),( 1 , 5 )
,( 1 , 6 ),( 2 , 1 ),( 2 , 2 ),( 2 , 3 ),( 2 , 4 ),( 2 , 5 ),( 2 , 6 ),( 3 , 1 ),( 3 , 2 )
,( 3 , 3 ),( 3 , 4 ),( 3 , 5 ),( 3 , 6 ),( 4 , 1 ),( 4 , 2 ),( 4 , 3 ),( 4 , 4 ),( 4 , 5 )
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
1
1
1
1
1
1
H
or
T
P ( not A ) = 1 - P ( A )
P ( A or B ) = P ( A ) + P ( B ) ( mutually exclusive outcomes )
P ( A and B ) = P ( a ) x P ( B ) ( independent outcomes )
Blue die
Or
1st die
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,( 4 , 6 ),( 5 , 1 ),( 5 , 2 ),( 5 , 3 ),( 5 , 4 ),( 5 , 5 ),( 5 , 6 ),( 6 , 1 ),( 6 , 2 )
,( 6 , 3 ),( 6 , 4 ),( 6 , 5 ),( 6 , 6 ) }
Number of possible outcome = 36
( 1 ) = P ( 5 ) = ? ( ႏစခေပါငး 5 ရမည ျဖစတနစြမး )
The set of favourable outcomes = { ( 1,4 ),( 2,3 ),( 3,2 ),( 4,1 ) }
Number of favourable outcomes = 4
P ( A ) = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑁𝑢𝑚𝑏 𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
P ( 5 ) = 4
36 =
1
9
---------------------------------------------------------------------
( 2 ) P ( not 5 ) = ?
P ( not 5 ) = 1 – P ( 5 ) = 1 - 1
9 =
8
9
---------------------------------------------------------------------
( 3 ) P ( 5 or 10 ) = ?
ႏစခေပါငး 5 ကသည ျဖစတနစြမး ( သ႔ ) 10 ကသည ျဖစတနစြမး
The set of all possible outcomes = { ( 1,4 ),( 2,3 ),( 3,2 ),( 4,1 ),( 4,6 ),( 5,5 ),( 6,4 ) }
Number of possible outcomes = 7
P ( 5 or 10 ) = 7
36
Mutually exclusive formula အရ တြကမညဆလင
P ( 5 or 10 ) = P ( 5 ) + P ( 10 )
The set of all possible outcomes = { ( 4,6 ),( 5,5 ),( 6,4 ) }
Number of possible outcomes = 3
P ( 10 ) = 3
36
P ( 5 or 10 ) = P ( 5 ) + P ( 10 ) = 4
36 +
3
36 =
7
36
---------------------------------------------------------------------
( 4 )P ( Blue 4 ) = ?
အျပာေရာငအစာတ 4 ကမည ျဖစတနစြမး = ( 4,1 ),( 4,2 ),( 4,3 ),( 4,4 ) ,( 4,5 ),( 4,6 )
The set of all possible outcomes = { ( 4,1 ),( 4,2 ),( 4,3 ),( 4,4 ) ,( 4,5 ),( 4,6 ) }
Number of possible outcomes = 6
P ( Blue 4 ) = 6
36 =
1
6
---------------------------------------------------------------------
( 5 )P ( Black 5 ) = ?
အနကေရာငအစာတ 4 ကမည ျဖစတနစြမး = ( 1,5 ),( 2,5 ),( 3,5 ),( 4,5 ) ,( 5,5 ),( 6,5 )
The set of all possible outcomes = { ( 1,5 ),( 2,5 ),( 3,5 ),( 4,5 ) ,( 5,5 ),( 6,5 ) }
Number of possible outcomes = 6
P ( Black 5 ) = 6
36 =
1
6
---------------------------------------------------------------------
( 6 ) P ( Blue 4 and Black 5 ) = ?
အျပာေရာငအစာတ 4 ႏင အနကေရာငအစာတ 4 တၿပငတညး ကမည ျဖစတနစြမး =
The set of all possible outcomes = { ( 4,5 ) }
Number of possible outcomes = 1
P (Blue 4 and Black 5 ) = 1
36
Independent event formula အရ
P ( Blue 4 and Black 5 ) = P ( Blue 4 ) x P (Black 5 ) = 1
6 x
1
6 =
1
36
မတခက။ ။Total score ေမးလင အနညးဆး 2 မ အမားဆး 12 အထ ရနင။
အစာတးတစတးခငးေပၚရ တနဖးေမးလင အနညးဆး 1 မ အမားဆး 6 အထ ရနင။
ႏစခေပါငး 5 ကသည ျဖစတနစြမး = ( 1,4 ),( 2,3 ),( 3,2 ),( 4,1 )
ႏစခေပါငး 10 ကသည ျဖစတနစြမး = ( 4,6 ),( 5,5 ),( 6,4 )
ႏစခေပါငး 5 ကသည ျဖစတနစြမး ( သ႔ ) 10 ကသည ျဖစတနစြမး
= ( 1,4 ),( 2,3 ),( 3,2 ),( 4,1 ) ,( 4,6 ),( 5,5 ),( 6,4 )
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Exercise 7.2
( 7 ) ( a ) P ( a 1 on the blue dice ) = ?
black dice ဘာျဖစျဖစ blue dice 1 ကမည ျဖစရပ
Number of possible outcomes ={ ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 ) }
The set of all possible outcomes = 6
P ( a 1 on the blue dice ) = 6
36 =
1
6
---------------------------------------------------------------------
( b ) P ( a 1 on the blue dice or a 6 on the blue dice ) = P ( blue 1 or blue 6 ) = ?
black dice ဘာျဖစျဖစ blue dice 1 ကမည ျဖစရပ (သ )
black dice ဘာျဖစျဖစ blue dice 6 ကမည ျဖစရပ
The set of favourable outcomes = { ( 1 ,1 ),( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 ),( 6,1 )
,( 6,2 ),( 6,3 ),( 6,4 ),( 6,5 ),( 6,6 ) }
Number of favourable outcomes = 12
P ( blue 1 or blue 6 ) = 12
36 =
1
3
---------------------------------------------------------------------
( c ) P ( a 1 on the blue dice or a 5 on the black dice ) = P ( blue 1 or black 5 ) =?
black dice ဘာျဖစျဖစ blue dice 1 ကမည ျဖစရပ (သ )
blue dice ဘာျဖစျဖစ black dice 5 ကမည ျဖစရပ
The set of favourable outcomes = { ( 1 ,1 ),( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 ),( 1,5 )
,( 2,5 ),( 3,5 ),( 4,5 ),( 5,5 ),( 6,5 ) }
Number of favourable outcomes = 12
P ( blue 1 or black 6 ) = 12
36 =
1
3
---------------------------------------------------------------------
( 9 ) Eg ( 1 ) - 4,5,6 ႏငအတတပင။ ပထမေမးခြနးအတြကတြရာေပး ၊ ဒတယေမးခြနးက
သးခငးတာျဖစ၍ ခြရာေပး။
( 10 ) ပထမေမးခြနးမာ Independent formula မသးပ ရာခငးျခငးျဖစသည။
P ( blue 1 and black number greater than 4 ) =
P ( blue 1 and black number > 4 ) = ?
blue die 1 ႏင Black die 4 ထကႀကးသည တနဖးကမည ျဖစရပ
The set of favourable outcomes = { ( 1,5 ),( 1,6 ) }
Number of favourable outcomes = 2
P ( blue 1 and black number > 4 ) = 2
36 =
1
18
ဒတယေမးခြနးမာ ထေမးခြနးကပင Independent formula သး၍ ရာခငးျခငးျဖစသည။
P ( blue 1 and black number > 4 ) = P ( blue 1 ) + P ( black number > 4 )
P ( blue 1 ) = ?
Black die ဘာျဖစျဖစ blue die 1 ကမည ျဖစရပ
blue dice black dice The set of favourable outcomes
1 1,2,3,4,5,6 ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 )
blue dice black dice The set of favourable outcomes
1 1,2,3,4,5,6 ( 1,1 ),( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 )
( or )
blue dice black dice The set of favourable outcomes
6 1,2,3,4,5,6 ( 6,1 ),( 6,2 ),( 6,3 ),( 6,4 ),( 6,5 ),( 6,6 )
blue dice black dice The set of favourable outcomes
1 1,2,3,4,5,6 ( 1,1 ), ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 )
( or )
blue dice black dice The set of favourable outcomes
1,2,3,4,5,6 5 ( 1,5 ),( 2,5 ),( 3,5 ),( 4,5 ),( 5,5 ),( 6,5 )
blue dice black dice The set of favourable outcomes
1 5,6 ( 1,5 ),( 1,6 )
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Number of possible outcomes = { ( 1,1 ), ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 ) }
The set of all possible outcomes = 6
P ( blue 1 ) = 6
36 =
1
6
---------------------------------------------------------------------
P ( black number > 4 ) = ?
blue dice ဘာျဖစျဖစ black dice 4 ထကႀကးရမည ျဖစရပ
Number of possible outcomes = { ( 1,6 ),( 2,6 ),( 3,6 ),( 4,6 ),( 5,6 ),( 6,6 )
( 1,5 ),( 2,5 ),( 3,5 ),( 4,5 ),( 5,5 ),( 6,5 ) }
The set of all possible outcomes = 12
P ( black number > 4 ) = 12
36 =
1
3
P ( blue 1 and black number > 4 ) = P ( blue 1 ) + P ( black number > 4 )
= 1
6 +
1
3 =
1
18
-----------------------------------------------------------------
The set of favourable outcomes Total Score
Even or
Odd
Total Score
Is Prime
No of Outcome
P ( A ) value
( 1,1 ) 2 E Prime 1 P ( 2 ) = 1
36
( 1,2 ),( 2,1 ) 3 O Prime 2 P ( 3 ) = 1
18
( 1,3 ),( 2,2 ),( 3,1 ) 4 E Not Prime 3 P ( 4 ) = 1
12
( 1,4 ),( 2,3 ),( 3,2 ),( 4,1 ) 5 O Prime 4 P ( 5 ) = 1
9
( 1,5 ),( 2,4 ),( 3,3 ),( 4,2 ), ( 5,1 ) 6 E Not Prime 5 P ( 6 ) = 5
36
( 1,6 ),( 2,5 ),( 3,4 ),( 4,3 ), ( 5,2 ), ( 6,1 ) 7 O Prime 6 P ( 7 ) = 1
6
( 2,6 ),( 3,5 ),( 4,4 ), ( 5,3 ), ( 6,2 ) 8 E Not Prime 5 P ( 8 ) = 5
36
( 3,6 ),( 4,5 ), ( 5,4 ), ( 6,3 ) 9 O Not Prime 4 P ( 9 ) = 1
9
( 4,6 ), ( 5,5 ), ( 6,4 ) 10 E Not Prime 3 P ( 10 ) = 1
12
( 5,6 ), ( 6,5 ) 11 O Prime 2 P ( 11 ) = 1
18
( 6,6 ) 12 E Not Prime 1 P ( 12 ) = 1
36
( 1 ) Probability of total score of 2 = P ( 2 ) = ?
The set of favourable outcomes = { ( 1,1 ) }
Number of favourable outcomes = 1
P ( 2 ) = 1
36
---------------------------------------------------------------------
P ( 3 ) = ?
The set of favourable outcomes = { ( 1 , 2 ) , ( 2 , 1 ) }
Number of favourable outcomes = 2
P ( 3 ) = 2
36 =
1
18
---------------------------------------------------------------------
P ( 4 ) = ?
blue dice black dice The set of favourable outcomes
1 1,2,3,4,5,6 ( 1,1 ), ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 )
blue dice black dice The set of favourable outcomes
1,2,3,4,5,6 5,6 ( 1,5 ),( 2,5 ),( 3,5 ),( 4,5 ),( 5,5 ),( 6,5 )
( 1,6 ),( 2,6 ),( 3,6 ),( 4,6 ),( 5,6 ),( 6,6 )
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The set of favourable outcomes = { ( 1 , 3 ) , ( 2 , 2 ), ( 3 , 1 ) }
Number of favourable outcomes = 3
P ( 4 ) = 3
36 =
1
12
---------------------------------------------------------------------
P ( 5 ) = ?
The set of favourable outcomes = { ( 1 , 4 ) , ( 2 , 3 ), ( 3 , 2 ) , ( 4 , 1 ) }
Number of favourable outcomes = 4
P ( 5 ) = 4
36 =
1
9
---------------------------------------------------------------------
P ( 6 ) = ?
The set of favourable outcomes = { ( 1 , 5 ) , ( 2 , 4 ), ( 3 , 3 ) , ( 4 , 2 ) ,( 5 , 1 ) }
Number of favourable outcomes = 5
P ( 6 ) = 5
36
---------------------------------------------------------------------
P ( 7 ) = ?
The set of favourable outcomes = {( 1 , 6 ),( 2 , 5 ),( 3 , 4 ),( 4 , 3 ),( 5 , 2 ),( 6 , 1 )}
Number of favourable outcomes = 6
P ( 7 )
---------------------------------------------------------------------
P ( 8 ) = ?
The set of favourable outcomes = {( 2 , 6 ),( 3 , 5 ),( 4 , 4 ),( 5 , 3 ),( 6 , 2 ) }
Number of favourable outcomes = 5
P ( 8 ) = 5
36
---------------------------------------------------------------------
P ( 9 ) = ?
The set of favourable outcomes = { ( 3 , 6 ),( 4 , 5 ),( 5 , 4 ),( 6 , 3 ) }
Number of favourable outcomes = 4
P ( 9 ) = 4
36 =
1
9
P ( 10 ) = ?
The set of favourable outcomes = { ( 4 , 6 ),( 5 , 5 ),( 6 , 4 ) }
Number of favourable outcomes = 3
P ( 10 ) = 3
36 =
1
12
---------------------------------------------------------------------
P ( 11 ) = ?
The set of favourable outcomes = { ( 5 , 6 ),( 6 , 5 ) }
Number of favourable outcomes = 2
P ( 11 ) = 2
36 =
1
18
---------------------------------------------------------------------
P ( 12 ) = ?
The set of favourable outcomes = { ( 6 , 6 ) }
Number of favourable outcomes = 1
P ( 11 ) = 1
36
---------------------------------------------------------------------
Are all these outcomes are equally likely ?
အထကပါ outcome မားသည တနဖးတ outcome မားျဖစၾကပါသလား။
outcome တစခခငး၏ Probability တနဖးမား တမတက ေမးခငးျဖစသည။
No , they are not equally likely outcomes.
---------------------------------------------------------------------
( 2 ) the most likeiy score = ?
ႀကမေရအမားဆးျဖစေပၚေသာ ေပါငးလဒ တနဖး ( သ ) Probability တနဖးအမားဆး ရေစသည ေပါငးလဒ
the most likeiy score = 7
the least likeiy score = ?
ႀကမေရအနညးဆးျဖစေပၚေသာ ေပါငးလဒ တနဖး ( သ ) Probability တနဖးအနညးဆး ရေစသညေပါငးလဒ
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the least likeiy score = 2 , 12
( 3 ) P ( total score is 2 or 12 ) = P ( 2 or 12 ) =?
P ( 2 or 12 ) = P ( 2 ) + P ( 12 ) = 1
36 +
1
36 =
2
36 =
1
18
2nd method
( စစေပါငးရလဒ 2 ( သ ) 12 ရေစမည ျဖစရပ = { ( 1 , 1 ) , ( 6 , 6 ) }
The set of favourable outcomes = { ( 1 , 1 ) , ( 6 , 6 ) }
Number of favourable outcomes = 2
P ( 2 or 12 ) = 2
36
ဟတြကလညး ရပါသည။ သ ေသာ အေပၚတြင ျဖစရပတစခခငးစအတြက Probability တနဖးမား
ရာျပသားရသျဖင ၄ငးတ အား အဆငသငသး၍ ခြရာေပးျခငးျဖစပါသည။ )
---------------------------------------------------------------------
( 4 ) P ( total score is 3 or 4 or 5 ) = P ( 3 or 4 or 5 ) =?
P ( 3 or 4 or 5 ) = P ( 3 ) + P ( 4 ) + P ( 5 ) = 2
36 +
3
36 +
4
36 =
6
36 =
1
4
---------------------------------------------------------------------
( 5 ) P ( total score is prime number ) = ? ( 2012 , ကခင ) ( 2013 , ရခင )
ႏစခေပါငးရလဒသည သဒၵကနးျဖစရမညဟဆသျဖင 2 ( သ ) 3 ( သ ) 5 ( သ ) 7 ( သ ) 11 တစခခ
ျဖစရပါမည။
P (total score is prime number ) = P ( 2 or 3 or 5 or 7 or 11 )
= P ( 2 ) + P ( 3 ) + P ( 5 ) + P ( 7 ) + P ( 11 ) = 1
36 +
3
36 +
4
36 +
6
36 +
2
36
= 15
36 =
5
12 ( 2014 , မြန၊ကရငတနသၤာရ )
---------------------------------------------------------------------
( 6 ) P ( total score is greater than 7 ) = P ( total score > 7 ) = ?
ႏစခေပါငးရလဒသည 7 ထကႀကးရမညဟဆသျဖင 8 ( သ ) 9 ( သ ) 10 ( သ ) 11 ( သ ) 12
တစခချဖစရပါမည။
P ( total score > 7 ) = P ( 8 or 9 or 10 or 11 or 12 )
= P ( 8 ) + P ( 9 ) + P ( 10 ) + P ( 11 ) + P ( 12 ) = 5
36 +
4
36 +
3
36 +
2
36 +
1
36
= 15
36 =
5
12
---------------------------------------------------------------------
( 8 ) P ( score on 2nd dice is greater than score on 1st dice ) =
P ( score on 2nd dice > score on 1st dice ) = ? ( 2014 , မြန၊ကရငတနသၤာရ )
The set of favourable outcomes = { ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 ),( 2,3 ),( 2,4 ),( 2,5 )
,( 2,6 ),( 3,4 ),( 3,5 ),( 3,6 ),( 4,5 ),( 4,6 ),( 5,6 ) }
Number of favourable outcomes = 15
P ( score on 2nd dice > score on 1st dice ) = 15
36 =
5
12
---------------------------------------------------------------------
1 mark
( 1 ) ( 2014 , Ygn ) ( 2007 , ပခး ) ( 2005 , မႏ ေလး )
poerfect square = ႏစထပကနးတမား
1 ၏ႏစထပကနးတ = 12 = 1 , 2 ၏ႏစထပကနးတ = 22
= 4 , 3 ၏ႏစထပကနးတ = 32 = 9
1st dice 2nd dice The set of favourable outcomes
1 2,3,4,5,6 ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 )
2 3,4,5,6 ( 2,3 ),( 2,4 ),( 2,5 ),( 2,6 )
3 4,5,6 ( 3,4 ),( 3,5 ),( 3,6 )
4 5,5 ( 4,5 ),( 4,6 )
5 6 ( 5,6 )
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The set of favourable outcomes = { ( 1,3 ),( 2,2 ),( 3,1 ),( 3,6 ),( 4,5 ),( 5,4 ),( 6,3 ) }
Number of favourable outcomes = 7
P ( total score will be poerfect square ) = 7
36
---------------------------------------------------------------------
( 2 ) ( 2003 , Ygn )
P ( total score will not be poerfect square )
= 1 - P ( total score will be poerfect square )
= 1 - 7
36 =
29
36
---------------------------------------------------------------------
( 3 ) ( 2013 , ႏငငျခား )
P ( total score will be divisible by 4 ) = ?
The set of favourable outcomes = { ( 1,3 ),( 2,2 ),( 3,1 ),( 2,6 ),( 3,5 ),( 4,4 )
,( 5,3 ),( 6,2 ),( 6,6 ) }
Number of favourable outcomes = 9
P ( total score will be divisible by 4 ) = 9
36 =
1
4
---------------------------------------------------------------------
( 4 ) ( 2012 , Ygn )
P ( Blue 4 or Black 4 )
= P (( 4,1 ),( 4,2 ),( 4,3 ),( 4,4 ),( 4,5 ),( 4,6 ), ( 1,4 ),( 2,4 ),( 3,4 ),( 5,4 ),( 6,4 ))
= 11
36
---------------------------------------------------------------------
( 5 ) ( 2007 , စစကငး ။ခငး ) ( 2013 , Ygn )
P ( total score will be even number ) = 𝟏𝟖
𝟑𝟔 =
𝟏
𝟐
( 6 ) ( 2007 , ႏငငျခား )
The set of favourable outcomes = ( 3 , 2 ),( 6 , 2 ),( 3 , 4 ),( 6 , 4 ),( 3 , 6 ),( 6 , 6 )
Number of favourable outcomes = 6
P ( multiple of 3 on 1st die and multiple of 3 on 2nd die ) = 6
36 =
1
6
( or )
2nd Method
P ( multiple of 3 on 1st die and multiple of 3 on 2nd die )
= P ( 3 0r 6 on 1st die and 2 or 4 or 6 on 2nd die )
= 6
36 + 6
36 𝐱 6
36 + 6
36 + 6
36 = 1236 𝐱
1836 =
1
3 𝐱
1
2 = =
1
6
---------------------------------------------------------------------
perfect square The set of favourable outcomes ျဖစေနသညေပါငးလဒမား
4 ( 1,3 ), ( 2,2 ),( 3,1 ),
9 ( 3,6 ),( 4,5 ),( 5,4 ),( 6,3 )
4 ျဖငစား၍ျပတသည The set of favourable outcomes ျဖစေနသညေပါငးလဒမား
4 ( 1,3 ), ( 2,2 ),( 3,1 )
8 ( 2,6 ),( 3,5 ),( 5,3 ),( 6,2 )
12 ( 6,6 )
multiple of 3 on 1st die and multiple of 3 on 2nd die
1st dice 2nd dice The set of favourable outcomes
3,6 2,4,6 ( 3,2 ),( 3,2 ),( 3,6 ),( 6,2 ),( 6,4 ),( 6,6 )
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( 8 ) total score = x , x < 7 , P ( x ) = = 1
9 =
4
36 ( 2012 , မြန၊ကရငတနသၤာရ )
The set of favourable outcomes Total Score
Total Score is Even or Odd
Total Score Is Prime
( 1,1 ) 2 E Prime
( 1,2 ),( 2,1 ) 3 O Prime
( 1,3 ),( 2,2 ),( 3,1 ) 4 E Not Prime
( 1,4 ),( 2,3 ),( 3,2 ),( 4,1 ) 5 O Prime
( 1,5 ),( 2,4 ),( 3,3 ),( 4,2 ), ( 5,1 ) 6 E Not Prime
( 1,6 ),( 2,5 ),( 3,4 ),( 4,3 ), ( 5,2 ), ( 6,1 ) 7 O Prime
( 2,6 ),( 3,5 ),( 4,4 ), ( 5,3 ), ( 6,2 ) 8 E Not Prime
( 3,6 ),( 4,5 ), ( 5,4 ), ( 6,3 ) 9 O Not Prime
( 4,6 ), ( 5,5 ), ( 6,4 ) 10 E Not Prime
( 5,6 ), ( 6,5 ) 11 O Prime
( 6,6 ) 12 E Not Prime
x = 5
---------------------------------------------------------------------
( 9 ) total score = x , x > 6 , P ( x ) = = 1
6 ( 2009 , ႏငငျခား )
x = ? Ans ;
---------------------------------------------------------------------
( 10 ) total score = x , x > 5 , P ( x ) = = 1
12 ( 2005 , ရမး၊ကယား )
x = ?
---------------------------------------------------------------------
( 11 ) P ( both die are even ) =
P ( 1st die is even and 2nd die is odd ) =
Old Question ( 5marks )
( 2014 , ရခင )
P ( total score is a multiple of 3 ) = ?
P ( the product of the score is divisible by 4 ) = ?
The set of all possible outcome = { ( 1 , 1 ), ( 1 , 2 ), ( 1 , 3 ),( 1 , 4 ),( 1 , 5 )
,( 1 , 6 ),( 2 , 1 ),( 2 , 2 ),( 2 , 3 ),( 2 , 4 ),( 2 , 5 ),( 2 , 6 ),( 3 , 1 ),( 3 , 2 )
,( 3 , 3 ),( 3 , 4 ),( 3 , 5 ),( 3 , 6 ),( 4 , 1 ),( 4 , 2 ),( 4 , 3 ),( 4 , 4 ),( 4 , 5 )
,( 4 , 6 ),( 5 , 1 ),( 5 , 2 ),( 5 , 3 ),( 5 , 4 ),( 5 , 5 ),( 5 , 6 ),( 6 , 1 ),( 6 , 2 )
,( 6 , 3 ),( 6 , 4 ),( 6 , 5 ),( 6 , 6 ) }
Number of possible outcome = 36
P ( A ) = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
The set of favourable outcomes = { ( 1,2 ), ( 2,1 ),( 1,5 ),( 2,4 ),( 3,3 ),( 4,2 )
,( 5,1 ),( 3,6 ), ( 4,5 ),( 5,4 ),( 6,3 ),( 6,6 ) }
Number of favourable outcomes = 12
P ( total score is a multiple of 3 ) = 12
36 =
1
3
Multiple of 3 = 3,6,9,12
Multiple of 3 possible outcome ျဖစေနသညေပါငးလဒမား
3 ( 1,2 ), ( 2,1 )
6 ( 1,5 ),( 2,4 ),( 3,3 ),( 4,2 ) ,( 5,1 )
9 ( 3,6 ), ( 4,5 ),( 5,4 ),( 6,3 )
12 ( 6,6 )
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Black die ( or ) 2nd die
Number of favourable outcomes = { ( 1 , 4 ),( 2 , 2 ),( 2 , 4 ),( 2 , 6 ),( 3 , 4 )
,( 4 , 1 ),( 4 , 2 ),( 4 , 3 ),( 4 , 4 ),( 4 , 5 )
,( 4 , 6 ),( 5 , 4 ),( 6 , 2 ),( 6 , 4 ),( 6 , 6 ) }
The set of favourable outcomes = 15
P ( the product of the score is divisible by 4 ) = = 15
36 =
5
12
( 2013 , မႏ ေလး )
( 1 ) table ဆြ
( 2 ) The set of all possible outcomes ေရး
( 3 ) Number of possible outcomes ေရး
P ( the sum of the scores is odd ) = ?
The set of favourable outcomes = { ( 1 , 2 ),( 1 , 4 ),( 1 , 6 ),( 2 , 1 ),( 2 , 3 )
,( 2 , 5 ),( 3 , 2 ),( 3 , 4 ),( 3 , 6 ),( 4 , 1 )
,( 4 , 3 ),( 4 , 5 ),( 5 , 2 ),( 5 , 4 ),( 5 , 6 )
,( 6 , 1 ),( 6 , 3 ),( 6 , 5 ) }
Number of favourable outcomes = 18
P ( the sum of the scores is odd ) = 18
36 =
1
2
Black die ( or ) 2nd die
P ( the product of the scores is greater than 15 ) = ?
The set of favourable outcomes = { ( 3,6 ),( 4,4 ),( 4,5 ),( 4,6 ) ,( 5,4 ),( 5,5 )
,( 5,6 ),( 6,3 ),( 6,4 ),( 6,5 ),( 6,6 ) }
Number of favourable outcomes = 11
P ( the product of the scores > 15 ) = = 11
36
Black die ( or ) 2nd die
P ( the product of the score is multiple of 6 ) = ?
Multiple of 6 6,12,18,24,30,36
The set of favourable outcomes = { ( 6 , 1 ),( 6 , 2 ),( 6 , 3 ),( 6 , 4 ),( 6 , 5 ),( 6 , 6 )
,( 2 , 3 ),( 2 , 6 ),( 3 , 2 ),( 3 , 4 ),( 3 , 6 ),( 4 , 3 )
,( 4 , 6 ),( 5 , 6 ),( 1 , 6 ) }
Number of favourable outcomes = 15
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
Blue die
Or
1st die
Blue die
Or
1st die
Blue die
Or
1st die
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P ( the product of the score is multiple of 6 ) = = 15
36 =
5
12
---------------------------------------------------------------------
( 2013 , မြန၊ကရငတနသၤာရ )
P ( the sum of the scores is even ) = 1
2 ( ရာၿပး )
P ( the product of the scores is greater than 20 ) = = 1
6
P ( the product of the score is multiple of 6 ) = 5
12 ( ရာၿပး )
( 2011 , Ygn )
P ( the sum of the scores is less than 7 ) = ?
Black die ( or ) 2nd die
Ans ; P ( the sum of the scores is less than 7 ) = 15
36 =
5
12
P ( the product of the scores is even ) = ?
Black die ( or ) 2nd die
The set of favourable outcomes = { ( 1 , 2 ),( 1 , 4 ),( 1 , 6 ),( 2 , 1 ),( 2 , 2 ),( 2 , 3 )
,( 2 , 4 ),( 2 , 5 ),( 2 , 6 ),( 3 , 2 ),( 3 , 4 ),( 3 , 6 )
,( 4 , 1 ),( 4 , 2 ),( 4 , 3 ),( 4 , 4 ),( 4 , 5 ),( 4 , 6 )
,( 5 , 2 ),( 5 , 4 ),( 5 , 6 ),( 6 , 1 ),( 6 , 2 ),( 6 , 3 )
,( 6 , 4 ),( 6 , 5 ),( 6 , 6 ) }
Number of favourable outcomes = 27
P ( the product of the scores is even ) = 27
36 =
3
4
( 2012 , ရခင )
P ( score on 2nd die is greater than score on 1st die ) = = 5
12 ( ရာၿပး )
**P ( score on one die is prime and score on other die is even ) = ?
Black die ( or ) 2nd die
The set of favourable outcomes = { ( 2 , 2 ) ( 2 , 3 ),( 2 , 4 ),( 2 , 6 ),( 2 , 5 ),( 3 , 2 )
,( 3 , 4 ),( 3 , 6 ),( 4 , 2 ),( 4 , 3 ),( 4 , 5 ),( 5 , 2 )
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
Blue die
Or
1st die
Blue die
Or
1st die
အထးသတထားရန
Event တနဖးမားရာရာတြင sum ႏင product က အထးသတထားပါ။
ပစာၦက sum ကရာခငးတာလား product ကရာခငးတာလား အထးဂရစကရနလပါသည။
Blue die
Or
1st die
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,( 5 , 4 ),( 5 , 6 ),( 6 , 5 ),( 6 , 3 ),( 6 , 2 ) }
Number of favourable outcomes = 17
**P ( score on one die is prime and score on other die is even ) = = 17
36
( 2010 , ကခင )
P ( total score is a multiple of 3 ) = 12
36 =
1
3 ( ရာၿပး )
P ( the product of the score is divisible by 4 ) = 15
36 =
5
12 ( ရာၿပး )
( 2010 , ရမး၊ကယား )
P ( total score is prime number ) = 15
36 =
5
12 ( ရာၿပး )
P ( the total score is greater than 10 ) = ?
Black die ( or ) 2nd die
The set of favourable outcomes = { ( 6 , 6 ),( 6 , 5 ),( 5 , 6 ) }
Number of favourable outcomes = 3
P ( the total score is greater than 10 ) = = 3
36 =
1
12
( 2010 , ႏငငျခား )
P ( getting a total of 10 or more
Black die ( or ) 2nd die
The set of favourable outcomes = { ( 4,6 ),( 5,5 ),( 5,6 ),( 6,4 ),( 6,5 ),( 6,6 )}
Number of favourable outcomes = 6
P ( getting a total of 10 or more ) = = 6
36 =
1
6
P ( both dice show the same number ) = ?
The set of favourable outcomes = { ( 1,1 ),( 2,2 ),( 3,3 ),( 4,4 ),( 5,5 ),( 6,6 ) }
Number of favourable outcomes = 6
P ( both dice show the same number ) = = 6
36 =
1
6
( 2009 , မႏ ေလး )
P ( the sum of the score is prime number ) = P ( total score is prime number ) = 5
12
( ရာၿပး )
P ( the product of the score is divisible by 6 or 9 ) = ?
Black die ( or ) 2nd die
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
Blue die
Or
1st die
Blue die
Or
1st die
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The set of favourable outcomes = { ( 1 , 6 ),( 2 , 3 ),( 2 , 6 ),( 6 , 1 ).( 6 , 2 ),( 6 , 3 )
( 6 , 4 ),( 6 , 5 ),( 6 , 6 ),( 3 , 6 ),( 4 , 3 ),( 4 , 6 )
( 5 , 6 ),( 3 , 2 ),( 3 , 3 ),( 3 , 4 ) }
Number of favourable outcomes = 16
P ( the product of the score is divisible by 6 or 9 ) = 16
36 =
4
9
( 2009 , စစကငး၊ခငး )
P ( total score is less than 7 ) = P ( total score < 7 ) = 5
12
P ( the total score is not divisible by 3 ) = ?
Black die ( or ) 2nd die
The set of favourable outcomes = { ( 1 , 1 ),( 1 , 2 ),( 1 , 4 ),( 1 , 5 ),( 2 , 1 ),( 2 , 2 )
,( 2 , 4 ),( 2 , 5 ),( 4 , 1 ),( 4 , 2 ),( 4 , 4 ),( 4 , 5 )
,( 5 , 1 ),( 5 , 2 ),( 5 , 4 ),( 5 , 5 ) }
Number of favourable outcomes = 16
P ( the total score is not divisible by 3 ) = = 16
36 =
4
9
( 2009 , ရခင )
P ( total score is greater than 5 ) = P ( total score > 5 ) = = 13
18
P ( the total score is divisible by ) = = 18
36 =
1
2
( 2006 , ပခး )
P ( score on 1st die is 2 less than score on 2nd die ) = ?
The set of favourable outcomes = { ( 1,3 ),( 2,4 ),( 3,5 ),( 4,6 ) }
Number of favourable outcomes = 4
P ( score on 1st die is 2 less than score on 2nd die ) = 4
36 =
1
9
( 2008 , ရမး၊ကယား )
In a game , two dice ( dice A and B ) are used. Die A has 2 blue faces and 4 white
faces. Die B has 4 blue faces and 2 red faces. Die A and B are throw together. Find the
probability that just one die show a blue face on top.
Let B1 , B
2 ,
W1 , W
2 , W
3
, W4 are six
faces of die A
And b1 , b
2 ,
b3 , b
4 , r
1 , r
2 are six faces of die B
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
Blue die
Or
1st die
Blue die
Or
1st die
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Die B
Just = exactly
just one die show a blue face on top = exactly one die show a blue face on top
The set of all possible outcome = { ( B1,b
1 ),( B
1,b
2 ),( B
1,r
1 ),( B
1,r
2 ),( B
1,r
3 )
,( B1,r
4 ),( B
2,b
1 ),( B
2 ,b
2 ),( B
2,r
1 ),( B
2,r
2 ),( B
2,r
3 ),( B
2,r
4 ),( B
3,b
1 )
,( B3
,b2 ),( B
3,r
1 ),( B
3,r
2 ),( B
3,r
3 ),( B
3,r
4 ),( B
4,b
1 ),( B
4,b
2 ),( B
4,r
1 )
,( B4,r
2 ),( B
4,r
3 ),( B
4,r
4 ),( W
1,b
1 ),( W
1,b
2 ),( W
1,r
1 ),( W
1,r
2 ),( W
1,r
3 )
,( W1,r
4 )m( W
2,b
1 ),( W
2,b
2 ),( W
2,r
1 ),( W
2,r
2 ),( W
2,r
3 ),( W
2,r
4 ) }
Number of possible outcome = 36
Both dice show blue.
Just one die show blue.
P ( just one die show a blue face on top ) = ?
The set of favourable outcomes = { ( B1,r
1 ),( B
1,r
2 ),( B
1,r
3 ),( B
1,r
4 ),( B
2,r
1 )
,( B2,r
2 ),( B
2,r
3 ),( B
2,r
4 ),( B
3,r
1 ),( B
3,r
2 )
,( B3,r
3 ),( B
3,r
4 ),( B
4,r
1 ),( B
4,r
2 ),( B
4,r
3 )
,( B4,r
4 ),( W
1,b
1 ),( W
1,b
2 ),( W
2,b
1 ),( W
2,b
2 )
Number of favourable outcomes = 20
P ( just one die show a blue face on top ) = 20
36 =
5
9
( or )
P ( just one die show a blue face on top ) = ?
P ( blue face on die A ) = 2
6 =
1
3
P ( white face on die A ) = P ( not blue face on die A ) = 4
6 =
2
3
P ( blue face on die B ) = 4
6 =
2
3
P ( white face on die B ) = P ( not blue face on die B ) = 2
6 =
1
3
P ( just one die show a blue face on top )
= P ( blue face on die A and not blue face on die B ( or ) not blue face on die A and
blue face on die B )
= P ( blue face on die A and not blue face on die B ) + P ( not blue face on die A and
blue face on die B )
= [ P ( blue face on die A ) x P ( not blue face on die B ) ]
+ [ P ( not blue face on die A ) x P ( blue face on die B ) ]
= [ 1
3 x
1
3 ] + [
2
3 x
2
3 ]
= 1
9 +
4
9
= 5
9
b1 b2 r1 r2 r3 r4
B1
B2
B3
B4
W1
W2
( B1,b1 ) ( B1,b2 ) ( B1,r1 ) ( B1,r2 ) ( B1,r3 ) ( B1,r4 )
( B2,b1 ) ( B2 ,b2 ) ( B2,r1 ) ( B2,r2 ) ( B2,r3 ) ( B2,r4 )
( B3,b1 ) ( B3 ,b2 ) ( B3,r1 ) ( B3,r2 ) ( B3,r3 ) ( B3,r4 )
( B4,b1 ) ( B4,b2 ) ( B4,r1 ) ( B4,r2 ) ( B4,r3 ) ( B4,r4 )
( W1,b1 ) ( W1,b2 ) ( W1,r1 ) ( W1,r2 ) ( W1,r3 ) ( W1,r4 )
( W2,b1 ) ( W2,b2 ) ( W2,r1 ) ( W2,r2 ) ( W2,r3 ) ( W2,r4 )
Die A
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ေနာကဆကတြေမးခြနးမား
( a )
P ( neither dice show a blue face on top ) ( သ႕မဟတ )
P ( neither die A nor Die B show a blue face on top ) = ?
P ( neither dice show a blue face on top )
= P ( both dice don’t show a blue face on top )
= P ( not blue face on die A and not blue face on die B )
= P ( not blue face on die A ) x P ( not blue face on die B )
= 2
3 x
1
3 =
2
9
အေပၚရ ဇယားကြကမတြကလင ၄ငးအေျခအေနမာ
P ( ( W1,r1 ),( W1,r2 ),( W1,r3 ),( W1,r4 ),( W1,r1 ),( W1,r2 ),( W1,r3 ),( W1,r4 ) ) = 8
36 =
2
9
( b )
P ( either die A or Die B show a blue face on top ) = ?
Since throwing two dice are mutually exclusive and indepewndent event
P ( either die A or Die B show a blue face on top )
= P ( just one die show a blue face on top ) = 20
36 =
5
9
( c )
P ( at least one die show a blue face on top ) = ?
P ( at least one die show a blue face on top )
= P ( blue face on die A and not blue face on die B ( or ) not blue face on die A and
blue face on die B ( or ) blue face on die A and blue face on die B)
= [ P ( blue face on die A ) x P ( not blue face on die B ) ]
+ [ P ( not blue face on die A ) x P ( blue face on die B ) ]
+ [ P ( blue face on die A ) x P ( blue face on die B ) ]
= [ 1
3 x
1
3 ] + [
2
3 x
2
3 ] + [
1
3 x
2
3 ]
= 1
9 +
4
9 +
2
9
= 7
9
အေပၚရ ဇယားကြကမတြကလင ၄ငးအေျခအေနမာ
P (( B1,r1 ),( B1,r2 ),( B1,r3 ) ,( B1,r4 ),( B2,b1 ),( B2 ,b2 ),( B2,r1 ),( B2,r2 ),( B2,r3 ),
( B2,r4 ),( B3,r1 ),( B3,r2 ),( B3,r3 ),( B3,r4 ),( B4,r1 ),( B4,r2 ),( B4,r3 ),( B4,r4 )
,( W1,b1 ),( W1,b2 ),( W1,r1 ),( W1,r2 ),( W1,r3 ),( W1,r4 )m( W2,b1 ),( W2,b2 ),( W2,r1 ),( W2,r2 )
,( W2,r3 ),( W2,r4 ) ) = 28
36 =
7
9
( d )
P ( at most one die show a blue face on top ) = ?
P ( at most one die show a blue face on top )
= P ( blue face on die A and not blue face on die B ( or ) not blue face on die A and
blue face on die B ( or ) not blue face on die A and not blue face on die B)
= [ P ( blue face on die A ) x P ( not blue face on die B ) ]
+ [ P ( not blue face on die A ) x P ( blue face on die B ) ]
+ [ P ( blue face on die A ) x P ( blue face on die B ) ]
= [ 1
3 x
1
3 ] + [
2
3 x
2
3 ] + [
1
3 x
2
3 ]
= 1
9 +
4
9 +
2
9
= 7
9
---------------------------------------------------------------------
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Table အတြက ေနာကဆကတြေမးခြနးမား
( 1 )
P ( at least one die show 3 ) = ?
Black die ( or ) 2nd die
P ( at least one die show 3 ) = 11
36
---------------------------------------------------------------------
( 2 )
P ( at most one die show 3 ) = ?
Black die ( or ) 2nd die
P ( at most one die show 3 ) = 35
36
---------------------------------------------------------------------
( 3 )P ( just one die show 3 ) = 10
36
Black die ( or ) 2nd die
---------------------------------------------------------------------
( 4 )P (either die A or die B show 3 ) = 10
36
Black die ( or ) 2nd die
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
Blue die
Or
1st die
Blue die
Or
1st die
Blue die
Or
1st die
Blue die
Or
1st die
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( 5 ) P ( exactly one show 3 ) = 10
36
Black die ( or ) 2nd die
---------------------------------------------------------------------
( 6 )
P ( neither die A nor die B show 3 ) = ?
Black die ( or ) 2nd die
P ( neither die A nor die B show 3 ) = 25
36
---------------------------------------------------------------------
( 7 )
P ( both dice show difference numbers ) = ?
Black die ( or ) 2nd die
P ( both dice show difference numbers )
= 1 - P ( both dice show same numbers ) = 1 - 6
36 =
30
36 =
5
6
---------------------------------------------------------------------
---------------------------------------------------------------------
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
1 2 3 4 5 6
1
2
3
4
5
6
( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )
( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )
( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )
( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )
( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )
( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )
ေပါငးလဒမား 2 3 4 5 6 7 8 9 10 11 12
( 1st , 2nd ) (blue,black )
( 1,1 ) ( 1,2 ) ( 1,3 ) ( 1,4 ) ( 1,5 ) ( 1,6 )
( 2,1 ) ( 2,2 ) ( 2,3 ) ( 2,4 ) ( 2,5 ) ( 2,6 )
( 3,1 ) ( 3,2 ) ( 3,3 ) ( 3,4 ) ( 3,5 ) ( 3,6 )
( 4,1 ) ( 4,2 ) ( 4,3 ) ( 4,4 ) ( 4,5 ) ( 4,6 )
( 5,1 ) ( 5,2 ) ( 5,3 ) ( 5,4 ) ( 5,5 ) ( 5,6 )
( 6,1 ) ( 6,2 ) ( 6,3 ) ( 6,4 ) ( 6,5 ) ( 6,6 )
Blue die
Or
1st die
Blue die
Or
1st die
Blue die
Or
1st die
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Tossing Coin Problems
Exercise ( 7.2 )
( 11 ) Copy and complete the table for the toss of a coin and the roll of a die.
2nd Coin
The set of all possible outcome = { ( H , H ),( H , T ),( T , H ),( T , T ) }
Number of possible outcome = 4
P ( H , H ) = P ( getting two heads ) = ? ( 2012 , ရနကန )
The set of favourable outcomes = { ( H , H ) }
Number of favourable outcomes = 1
P ( H , H ) = 1
4
P ( T , T ) = ?
The set of favourable outcomes = { ( T , T ) }
Number of favourable outcomes = 1
P ( T , T ) = 1
4
P ( a head and a tail in any order ) = ?
The set of favourable outcomes = { ( H , T ) , ( T , H ) }
Number of favourable outcomes = 2
P ( T , T ) = 2
4 =
1
2
ေနာကဆကတြေလကငရန
P ( at least one tail ) = ? ( 2015 , ျပညတြငး )
The set of favourable outcomes ={ ( H , H ), ( H , T ),( T , H ),( T , T ) }
Number of favourable outcomes = 3
P ( at least one tail ) = 3
4
---------------------------------------------------------------------
P ( at least one head ) = ? ( 2011 , ရနကန ) ( 2009 , ရနကန၊ မႏ ေလး )
The set of favourable outcomes = { ( H , H ),( H , T ),( T , H ) ,( T , T ) }
Number of favourable outcomes = 3
P ( at least one head ) = 3
4
---------------------------------------------------------------------
P ( at most one tail ) = ? ( 2013 , ပခး )
The set of favourable outcomes = { ( H , H ),( H , T ),( T , H ), ( T , T ) }
Number of favourable outcomes = 3
P ( at most one tail ) = 3
4
---------------------------------------------------------------------
P ( not getting two tails ) = ?
The set of favourable outcomes = { ( H , H ),( H , T ),( T , H ), ( T , T ) }
Number of favourable outcomes = 3
P (not getting two tails ) = 3
4
---------------------------------------------------------------------
P ( at most one head ) = ? ( 2010 , ႏငငျခား )
The set of favourable outcomes = { ( H ,
H ) ,( H , T ),( T , H ),( T , T ) }
H T
H ( H , H ) ( H , T )
T ( T , H ) ( T , T )
1st Coin
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Number of favourable outcomes =
P ( ( H , T ),( T , H ),( T , T ) ) = 3
4
---------------------------------------------------------------------
P ( not getting two heads ) = ?
The set of favourable outcomes = { ( H , H ) ,( H , T ),( T , H ),( T , T ) }
Number of favourable outcomes = 3
P ( not getting two heads ) = 3
4
---------------------------------------------------------------------
P ( exactly one head ) = P ({ ( H , H ), ( H , T ),( T , H ) ,( T , T ) } )
P ( just one head ) = P ({ ( H , H ), ( H , T ),( T , H ) ,( T , T ) } )
P ( exactly one tail ) = P ({ ( H , H ), ( H , T ),( T , H ) ,( T , T ) } )
P ( just one tail ) = P ({ ( H , H ), ( H , T ),( T , H ) ,( T , T ) } )
P ( ( H , T ),( T , H ) ) = 3
4 ( 2010 , ရမး၊ကယား )
---------------------------------------------------------------------
( 15 ) Copy and complete this array of any ordered triples for the possible outcomes
When 3 coins are tossed simultaneously;
( a ) P ( exactly two Heads ) = ?
The set of all possible outcome = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of possible outcome = 8
The set of favourable outcomes ={ HHT, HTH, THH }
Number of favourable outcomes = 3
8
( b ) P ( two Heads and a Tail in any order ) = ?
The set of favourable outcomes ={ HHT, HTH, THH }
Number of favourable outcomes = 3
8
( c ) P ( 3 Tails ) = ?
The set of favourable outcomes = { TTT }
Number of favourable outcomes = 1
P ( 3 Tails ) = 1
---------------------------------------------------------------------
ေနာကဆကတြေလကငရန
P ( exactly one Head ) = ?
The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of favourable outcomes = 3
P ( exactly one Head ) = 3
8
---------------------------------------------------------------------
P ( exactly one Tail ) = ?
The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of favourable outcomes = 3
P ( exactly one Tail ) = 3
8
---------------------------------------------------------------------
P ( at least one tail ) = ?
The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of favourable outcomes = 7
P ( at least one tail ) = 7
8
---------------------------------------------------------------------
P ( at least one head ) = ? ( 2013 , ကခင )
The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of favourable outcomes = 7
HHH HHT HTH HTT
THH THT TTH TTT
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P ( at least one head ) = 7
8
---------------------------------------------------------------------
P ( at most one tail ) = ?
The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of favourable outcomes = 4
P ( at most one tail ) = 4
8 =
1
2
---------------------------------------------------------------------
P ( at most one head ) = ?
The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of favourable outcomes = 4
P ( at most one head ) = 4
8 =
1
2
---------------------------------------------------------------------
P ( just one head ) = P ( exactly one head )
---------------------------------------------------------------------
P ( just one tail ) = P ( exactly one tail )
---------------------------------------------------------------------
P ( exactly two Heads ) = ?
The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of favourable outcomes = 3
P ( exactly one Head ) = 3
8
---------------------------------------------------------------------
P ( exactly two Tails ) = ? ( 2011 , မြနကရင၊တနသၤာရ )
The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of favourable outcomes = 3
P ( exactly one Tail ) = 3
8
---------------------------------------------------------------------
P ( at least two tails ) = ?
The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of favourable outcomes = 4
P ( at least one tail ) = 4
8 =
1
2
---------------------------------------------------------------------
P ( at least two heads ) = ? ( 2014 , ကခင )
The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of favourable outcomes =
P ( at least one head ) = 4
8 =
1
2
---------------------------------------------------------------------
P ( at most two tails ) = ?
The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of favourable outcomes = 7
P ( at most two tails ) = 7
8
---------------------------------------------------------------------
P ( at most two heads ) = ?
The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of favourable outcomes = 7
P ( at most one head ) = 7
8
---------------------------------------------------------------------
P ( just two heads ) = P ( exactly two heads )
---------------------------------------------------------------------
P ( just two tails ) = P ( exactly two tails )
---------------------------------------------------------------------
( 2016 , ႏငငျခား ) ( 2011 , မႏ ေလး မာ 6 ႀကမေမး )
Number of possible outcomes for tossing 5 fair coils = 2 x 2 x 2 x 2 x 2 = 32
---------------------------------------------------------------------
( 2014 , ပခး )
Number of possible outcomes for tossing a fair coils 3 times and rolling a die once
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= 2 x 2 x 2 x 6 = 48
---------------------------------------------------------------------
( 2017 , ျပညတြငး )
A coin is toss 2 times ,
P ( at least one tail ) = x – 2
P ( at least one tail ) = 3
4 ( ရာၿပး )
x – 2 = 3
4
x = 3
4 + 2
x = 3+8
4 =
11
4
( 2011 , ပခး )
3 coin are toss ,
P ( at least one head ) = 1 – x
x = ? Ans ; 1
8
---------------------------------------------------------------------
( 2008 , ႏငငျခား )
3 coin are toss ,
P ( exactly two tails and a head ) = ?
The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Number of favourable outcomes = 3
P ( exactly two tails and a head ) = 3
8
---------------------------------------------------------------------
5 marks
( 2012 , ကခင )
A coin is toss 4 times. Head or Tail is record each time. Draw a tree diagram..Find the
probabilities of exactly one tail and at least one tail .
1st Toss 2nd Toss 3rd Toss 4th Toss Possible Outcomes
H ( H , H , H , H )
T ( H , H , H , T )
H ( H , H , T , H )
T ( H , H , T , T )
H ( H , T , H , H )
T ( H , T , H , T )
H ( H , T , T , H )
T ( H , T , T , T )
H ( T , H , H , H )
T ( T , H , H , T )
H ( T , H , T , H )
T ( T , H , T , T )
H ( T , T , H , H )
T ( T , T , H , T )
H ( T , T , T , H )
T ( T , T , T , T )
The set of all possible outcome = { ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )
, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T , H , H , H ),( T , H , H , T )
,( T , H , T , H ),( T , H , T , T ),( T , T , H , H ),( T , T , H , T ),( T , T , T , H )
,( T , T , T , T ) }
Number of possible outcome = 16
H
T
H
T
H
T
H
H
T
H
T
H
T
T
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P ( exactly one tail ) = ?
The set of favourable outcomes ={ ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )
, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T , H , H , H ),( T , H , H , T )
,( T , H , T , H ),( T , H , T , T ),( T , T , H , H ),( T , T , H , T ),( T , T , T , H )
,( T , T , T , T ) }
Number of favourable outcomes = 3
P ( exactly one tail ) = 3
16
P ( at least one tail ) = ?
The set of favourable outcomes ={ ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )
, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T , H , H , H ),( T , H , H , T )
,( T , H , T , H ),( T , H , T , T ),( T , T , H , H ),( T , T , H , T ),( T , T , T , H )
,( T , T , T , T ) }
Number of favourable outcomes = 15
P ( at least one tail ) = 15
16
( 2007 , ရမး၊ကယား )
P ( at most one tail ) = ?
The set of favourable outcomes = { ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )
, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T , H , H , H ),( T , H , H , T )
,( T , H , T , H ),( T , H , T , T ),( T , T , H , H ),( T , T , H , T ),( T , T , T , H )
,( T , T , T , T ) }
Number of favourable outcomes = 4
P ( at most one tail ) = 4
16 =
1
4
( 2008 , ႏငငျခား )
P ( getting no tail ) = ?
The set of favourable outcomes ={ ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )
, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T , H , H , H ),( T , H , H , T )
,( T , H , T , H ),( T , H , T , T ),( T , T , H , H ),( T , T , H , T ),( T , T , T , H )
,( T , T , T , T ) }
Number of favourable outcomes = 1
P ( getting no tail ) = 1
16
ေနာကဆကတြေလကငရန
( 1 )P ( exactly one Head ) = ?
The set of favourable outcomes = { ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )
, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T,H,H,H ),( T,H,H,T ),( T,H,T,H )
,( T,H,T,T ),( T,T,H,H ),( T,T,H,T ),( T,T,T,H ),( T,T,T ,T ) }
( ဖယရမညတနဖးမားအား မမဘာသာျခစထတေပးပါ။ )
Number of favourable outcomes = .
P ( exactly one Head ) = 16
( 2 )P ( exactly two Head ) = ?
The set of favourable outcomes = { ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )
, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T,H,H,H ),( T,H,H,T ),( T,H,T,H )
,( T,H,T,T ),( T,T,H,H ),( T,T,H,T ),( T,T,T,H ),( T,T,T ,T ) }
( ဖယရမညတနဖးမားအား မမဘာသာျခစထတေပးပါ။ )
Number of favourable outcomes = .
P ( exactly two Head ) = 16
( 3 ) P ( exactly one Tail ) = ?
The set of favourable outcomes ={ ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )
, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T,H,H,H ),( T,H,H,T ),( T,H,T,H )
,( T,H,T,T ),( T,T,H,H ),( T,T,H,T ),( T,T,T,H ),( T,T,T ,T ) }
( ဖယရမညတနဖးမားအား မမဘာသာျခစထတေပးပါ။ )
လေသာတနဖးအားျဖညေပးပါ
လေသာတနဖးအားျဖညေပးပါ
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Number of favourable outcomes = .
P ( exactly one Tail ) = 16
( 4 ) P ( exactly two Tail ) = ?
The set of favourable outcomes = { ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )
, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T,H,H,H ),( T,H,H,T ),( T,H,T,H )
,( T,H,T,T ),( T,T,H,H ),( T,T,H,T ),( T,T,T,H ),( T,T,T ,T ) }
( ဖယရမညတနဖးမားအား မမဘာသာျခစထတေပးပါ။ )
Number of favourable outcomes = .
P ( exactly two Tail ) = 16
( 5 ) P ( getting no Head ) = ?
The set of favourable outcomes ={ ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )
, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T,H,H,H ),( T,H,H,T ),( T,H,T,H )
,( T,H,T,T ),( T,T,H,H ),( T,T,H,T ),( T,T,T,H ),( T,T,T ,T ) }
( ဖယရမညတနဖးမားအား မမဘာသာျခစထတေပးပါ။ )
Number of favourable outcomes = .
P ( getting no Head ) = 16
( 6 ) P ( just one Tail ) = ?
The set of favourable outcomes ={ ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )
, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T,H,H,H ),( T,H,H,T ),( T,H,T,H )
,( T,H,T,T ),( T,T,H,H ),( T,T,H,T ),( T,T,T,H ),( T,T,T ,T ) }
( ဖယရမညတနဖးမားအား မမဘာသာျခစထတေပးပါ။ )
Number of favourable outcomes = .
P ( just one Tail ) = 16
( 6 ) P ( just two Tail ) = ?
The set of favourable outcomes ={ ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )
, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T,H,H,H ),( T,H,H,T ),( T,H,T,H )
,( T,H,T,T ),( T,T,H,H ),( T,T,H,T ),( T,T,T,H ),( T,T,T ,T ) }
( ဖယရမညတနဖးမားအား မမဘာသာျခစထတေပးပါ။ )
Number of favourable outcomes = .
P ( just two Tail ) = 16
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( 2010 , မေကြး )
A coil is tossed and a die is throw. Head or Tail and a number turn up are recorded
each time. Draw a tree diagram and list possible outcomes. Find the probability that
Head and odd number turn up.
Tossing Co[n 2nd Thrown Possible Outcomes
1 ( H , 1 )
2 ( H , 2 )
3 ( H , 3 )
4 ( H , 4 )
5 ( H , 5 )
6 ( H , 6 )
1 ( T , 1 )
လေသာတနဖးအားျဖညေပးပါ
လေသာတနဖးအားျဖညေပးပါ
လေသာတနဖးအားျဖညေပးပါ
လေသာတနဖးအားျဖညေပးပါ
လေသာတနဖးအားျဖညေပးပါ
H
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2 ( T , 2 )
3 ( T , 3 )
4 ( T , 4 )
5 ( T , 5 )
6 ( T , 6 )
The set of all possible outcome = { ( H , 1 ),( H , 2 ),( H , 3 ),( H , 4 ),( H , 5 )
( H , 6 ),( T , 1 ),( T , 2 ),( T , 3 ),( T , 4 ),( T , 5 ),( T , 6 ) }
Number of possible outcome = 12
P ( Head and odd number turn up ) = ?
The set of all possible outcome = { ( H , 1 ),( H , 2 ),( H , 3 ),( H , 4 ),( H , 5 )
( H , 6 ),( T , 1 ),( T , 2 ),( T , 3 ),( T , 4 ),( T , 5 ),( T , 6 ) }
( ဖယရမညတနဖးမားအား မမဘာသာျခစထတေပးပါ။ )
Number of possible outcome =
P ( Head and odd number turn up ) = 12
Ans ; 1
4
( 2010 , ရခင )
P ( Head and even number turn up ) = ?
The set of all possible outcome = { ( H , 1 ),( H , 2 ),( H , 3 ),( H , 4 ),( H , 5 )
( H , 6 ),( T , 1 ),( T , 2 ),( T , 3 ),( T , 4 ),( T , 5 ),( T , 6 ) }
( ဖယရမညတနဖးမားအား မမဘာသာျခစထတေပးပါ။ )
Number of possible outcome =
P ( Head and even number turn up ) = 12
Ans ; 1
4
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T
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Eg 4. Three tennis players A,B and C play each other only once. The probability that
A will beat B is 1
3 , B will beat C is
2
5 and C will beat A is
2
7 . Calculate the
probability that C win both game.
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