7 probability.doc

8/16/2019 7 Probability.doc

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7. PROBABILITY

7.1.1 Sample Space and Events

Task A : Determine the number of possible outcomes in the sample space and the event by listing the outcomes.

1. An experiment is conducted by rolling a fair dice

once. If A is the event of getting an even score,

a! express, in set notation,

i! the sample space,

ii! the event A,

b! determine the number of possible outcomes in


ii! the event A.

Ans"er : a!i! # $ % &

a!ii! A $ % &

b!i! n#! $

b!ii! nA! $

'. In an experiment "here a fair coin is tossed t"ice,

(T represents an outcome "here the first toss

results in a )head* and the second toss results in a

)tail*. If + is the event of obtaining at least one

)tail*,



ii! the event +, b! determine the number of possible outcomes in


ii! the event +.

Ans"er : a!i! # $

a!ii! + $

b!i! n#! $

b!ii! n+! $

. A card is picked randomly from the follo"ing cards.

If - is the event of obtaining vo"el,



ii! the event -,

b! determine the number of possible outcomes in i! the sample space,

ii! the event -.

Ans"er : a!i! # $

a!ii! - $

b!i! n#! $

b!ii! n-! $

. In an experiment, t"o numbers are randomlychosen, the first from set A $ % , / , 0 and the

second from set + $ % ' , , 2&. If D is the event of

getting a total score at least 13,

a! express, in set notation, 45ote : x , y!

represents an outcome "here x ∈ A and y ∈ +


ii! the event D,

b! determine the number of possible outcomes in


ii! the event D.

Ans"er : a!i! # $

a!ii! D $

b!i! n#! $

b!ii! nD! $

Ans"ers: 1. # $ %1,',,,/,0&6 A $ %',,0&6 n#! $ 06 nA! $ '. # $ %((,(T,T(,TT&6 + $ %(T,T(,TT&6 n#! $ 6 n+! $ '. # $ %(,7,+,A,T&6 - $ %7,A&6 n#! $ /6 n-! $ ' . # $ %,'!,,!,,2!,/,'!,/,!,/,2!,0,'!,0,!,0,2!& 6 n#! $ 8

D $ %,2!,/,2!,0,!,0,2!& nD! $

Probability 1

( 7 + A T

a. 7xperiment is an activity that "e carry out to observe a result.

b. 9utcome is the result of an experiment that "e observed.

c. #ample space is set of all the possible outcomes or results! in an experiment

d. The sample space is denoted by the letter S .

e. 7vent is the set of all the possible outcomes or results! in the sample space that "e "ish to get.

f. An event can be denoted by any capital letters except S .

g. nS ! denotes the number of possible outcomes in the sample space.h n A denotes the number of ossible outcomes in the event A.


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Task +: Determine the number possible outcomes in the sample space and the event "ithout listing the outcomes.

1. A box contains black balls, '

green balls and / red balls. A

ball is dra"n at random fromthe box. If A represents the

event not getting the black

balls, determine a! n#!, b! nA!.

Ans"ers : a! n#! $

b! nA! $

'. A box contains cards of "hich

each card is "ritten "ith an

alphabet from the "ord)TA+A(A5*. A card is

dra"n at random from the box.

If + represents the event ofobtaining vo"el card, determine

a! n#!,

b! n+!.

Ans"ers : a! n#! $

b! n+! $

. The table sho"s a set of

numbers. A number is chosen at

random from the set.

13 11 1/

1; '' 18

10 '/ 12 If - represents the event of

choosing number that is multiple

of /, determine

a! n#!,

b! n-!.

Ans"ers : a! n#! $

b! n-! $

. There are 1 fans in a hall, each

controlled by a s"itch. It iskno"n that four fans are spoilt.

T"o s"itches are turned on at

random in an experiment.

Probability '


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7.1.2 Probabilit o! an Event

Task A : Determine the robability of #imple 7vents.

1. A bag contains ; red balls, /

green balls and 0 pink balls. A

ball is picked at random. ind

the probability that a red ball is

picked.

Ans"er :

E $ 7vent red ball is picked

n(R) = 7,

n(S) = 18

12

;

!

!!

=

=S n

Rn R P

'. A card is picked randomly from

a pack of cards "hich contains

red cards and 0 yello" cards.

ind the probability that a

yello" card is picked.

Ans"er :

Y $ 7vent of picking yello" card

n(Y) =

n(S) =

=

=!

!!

S n

Y nY P

/,

. A letter is selected at random

from the "ord )(F#I-#*.

ind the probability of selecting

letter S .

Ans"er :

A $ 7vent of selecting letter S

n A! $

nS ! $

=

=!

!!

S n

An A P

;'

. A fair dice is flipped once. Ghatis the probability of obtaining a

score greater than H

Ans"er :

B $ 7vent of obtaining a score

greater than .

n B! $

nS ! $

,

1

/. Identical cards numbered from'3 to 3, are put in a bag. A card

is dra"n randomly from the

bag. ind the probability that

the number on the card is a

prime number.

'1.

0. 7ach of the letters from the"ord )TA+A(A5* is "ritten

on identical cards and then put

in a box. If a card is dra"n at

random from the box, calculate

the probability that the card

dra"n is a vo"el.

2

,

Probability

a! The probability that an event A occurs is represented by P A!

b! +y classical definition, ( ) ( )

( )S n

An

S inoutcomesof number

Ainoutcomesof number A P == , "here ( ) 13 ≤≤ A P

c! If P A! $ 3, then the event is sure to fail or is a certain impossibility!.

d! If P A! $ 1. then the event is sure to succeed or is an absolute surety!e! The probability that an event A does not occur is represented by P A’ ! "here P A’ ! $ 1 P A!


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;. A box containing all the letters

of the "ord )E9+A+ICITF*

"ritten on identical cards. If

one card is dra"n at randomfrom the box, find the

probability that the card dra"n

sho"s a consonant.

11;

2. A card is dra"n at random from a

bag "hich contains '0 identical

cards "ritten "ith all the

different letters of the alphabet.ind the probability that the card

dra"n sho"s a letter from the

"ord )+A(A#A*.

1,'

8. A fair dice is rolled t"ice.

-alculate the probability of

getting numbers "ith a sum

more than 2.

12/

Task + : Determine the probability of events involving the idea of permutation and combination.

1. All the different three?digit numbers formed from

the digits ', , 0 and 8 "ithout repetition are"ritten in small pieces of papers and then put in a

hat.

a! Determine the total number of three?digit

numbers in the hat.

b! If one of the number is dra"n randomly fromthe hat, calculate the probability that the

number is less than 33.

a! ' b! .1

'. #ix?letter codes are formed by using all the letters in

the "ord )(IT=5


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7.2 Probabilit o! "#t#all E$cl#sive Events

Task A : Determine "hether events are mutually exclusive and exhaustive.

1. A fair coin is flipped t"ice. 7vents A and + are

defined belo" :

A $ 7vent "here both flips sho" )heads*

+ $ 7vent "here the t"o flips result in at least a

)tail*

#tate, "ith a reason, "hether events A and + are a!mutually exclusive events,

b! exhaustive events.

a! Fes, because J b! Fes, because J

'. A fair dice is rolled. 7vent - is the event )a score of

less than 0* and event D is the event )a score of more

than '*. #tate, "ith a reason, "hether events - and D

a! are mutually exclusive events,

b! are exhaustive events.

a! 5o, because J b! Fes, because J

. The sample space # is given by

# $ % x : 3 K x K '1, x is an integer &.

7vent L and F are defined as follo"s:

L $ % x : x are even integers &

F $ % x : x are integers "hich are multiple of &

+y finding L!, F! and L ∪ F!, determine

"hether events L and F

a! are mutually exclusive events, b! are exhaustive events.

a! 5o, because J b! 5o, because J

. The sample space, #, and the events M and G are

defined as follo"s :

# $ #et of code "ords formed from all the letters of

the "ord )A#* "ithout repetition

M $ #et of code "ords that begin and end "ith

consonants G $ #et of code "ords that begin "ith a vo"el and

end "ith a consonant.

+y finding M!, G! and M ∪ G!, determine

"hether events M and G

a! are mutually exclusive events,

b! are exhaustive events.

a! Fes, because J b! 5o, because J

Probability /

a! If events A and + are mutually exclusive, then only one or the other event can occur at time.

b! T"o events A and + are mutually exclusive events if one of the follo"ing conditions happens :

i! if A∩ + $ % & or A∩ + $ ∅ ! or

ii! if nA∩ +! $ 3 or

iii! if A∩ + ! $ 3 or

iv! if A∪ +! $ A! N +!c! T"o events A and + are exhaustive events if one of the follo"ing conditions happens :

i! if A∪ + $ #

ii! if nA∪ +! $ n#! or

iii! if A∪ + ! $ 1


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Task + : Determine the probability of the events belo".

1. A box contains / red, yello"

and green marbles. A marble

is dra"n randomly from the

box. ind the probability thatthe colour of the marble is

yello" or green.

1';

'. A card is picked randomly from

the cards belo".

ind the probability of getting a

card "ith a consonant or a letter

9.

.,

. 9n a rack, there are 0 #cience

books, 1' athematics books

and ' (istory books. A book is

to be chosen randomly from therack. -alculate the probability

of choosing a #cience book or a

athematics book.

138

. A number is chosen at random

from the set %1, ', J , '3&.

ind the probability that thenumber is a multiple of or a

multiple of ;.

'3;

/. A bag contains / blue pens, ' red

pens and 1 green pen. A pen is

picked at random from the bag.ind the probability of getting a

blue or a green pen.

.,

0.

A card is picked randomly fromthe cards above. ind the

probability of getting a card

"ith digit or digit /.

;/

;. 5ine cards are "ritten "ith the

letters of the "ord #TATI#TI-.

If a card is selected randomly,

find the probability that the card

has the letter T or a vo"el.

,'

2. uthu chooses a number

randomly from a set # "here

# $ % x : 13 ≤ x ≤ '3 , x is an

integer&.

ind the probability that the

number is a prime number or a

multiple of /.

11;

8. A fair dice is rolled once. ind

the probability that the dice

sho"s an odd number or a

number more than /.

,'

Probability 0

9 = T - 9 7 #

/ / 0


7/14

7.% Probabilit o! Independent Events

Task A : Determine the probability of the events belo".

1. A fair coin and a fair dice aretossed simultaneously.

Determine the probability of

obtaining a tail and a numbergreater than .

Ans"er :

A $ getting a tail

+ $ getting a number greater

than

'

1! = A P ,

,

1! = B P

0

1

,

1

'

1

!!

!!and

=

×=

×= ∩= B P A P B A P B A P

'. The probabilities of Thomas passing in the 7nglish test and

#cience test are,

' and

.

1

respectively. -alculate the

probability that he "ill pass in

both test.

01

. In a certain region, it is observedthat the probability that "ill rain

in a day is/

'. ind the

probability that it "ill rain for '

days.

'/.

. The probability that a defective

bulb produced from a factory is

;

'. If Oack buys t"o bulbs,

find the probability that he "ill

get t"o good bulbs.

/. The probability that a shooter

hits the target is/

,. or '

shots, find the probability that

he "ill fail to hit the target both

times.

0. There are 8 pens in a pencil box

of "hich / are red. If t"o pens

are chosen randomly, one by

one "ith replacement, find the

probability that both pens are

not red.

Probability ;

a! If events A and + are independent, then the outcome of event A does not affect the occurrence of the

outcome of event + and vice versa.

b! If A and + are t"o independent events, then A∩ +! $ A! × +!.

c! If A, + and - are three independent events, then A∩ + ∩ -! $ A! × +! × -!d! 7xamples of independent events A and +:

i! -ase 1 : 7xperiment : Tossing a fair coin and rolling a fair dice. A $ 7vent of getting )head*

+ $ 7vent of getting prime number

ii! -ase ' : 7xperiment : Eolling a fair dice t"ice.

A $ 7vent of getting even number from the first roll

+ $ 7vent of getting even number from the second roll

iii! -ase : 7xperiment : A coin is tossed and a card is picked

A $ 7vent of getting )tail*

+ $ 7vent of picking a diamond card

e! A tree diagram is very useful in helping us to ans"er Buestions that involve independent events


8/14

.8'/

'/.

2110

Task + : Determine probability by using Tree Diagram.1. Ali shoots t"o arro"s. The probability of Ali hitting

the bull*s eye "ith each shot is/

,. +y dra"ing a

tree diagram, find the probability of Ali hitting the

bull*s eye at least once.

Ans"er :

A $ event hitting the bull*s eye

AP $ event not hitting the bull*s eye

hit bull*s eye at least once!

$ AA! N AAP! N APA!

$

+

+

/

,

/

'

/

'

/

,

/

,

/

,

$'/

'1

'. There is a 3Q chance that Ceela "ill cycle to school

in a school day. +y dra"ing a tree diagram, find the

probability that out of any t"o school days, Ceela

"ill cycle to school in only one day.

Ans"er :

- $ event cycling to school

-P $ event not cycling to school

3.'

. A box contains red cards and green cards. T"ocards are dra"n at random, one after another

"ithout replacement. +y dra"ing a tree diagram,

find the probability that both the cards are of

different colour.

. +ox A contains 1 black marble and / "hite marbles.+ox + contains black marbles and "hite

marbles. A marble is picked at random from box A

and box +. +y dra"ing a tree diagram, find the

probability that both of the marbles are of the same

colour.

Probability 2

A

A&

A

AA&

A&

/,

/,

/,

/

'

/'

/'

!irst s'ot second s'ot

possible

o#tcomes

AA

AAP

A&A

A&A !irst da second da

possible

o#tcomes


9/14

;.

.'18

/ A bag contains / black marbles and "hite marbles.

T"o marbles are dra"n at random, one after another

from the bag "ithout replacement. +y dra"ing a tree

diagram, find the probability that both of the marbles

are of the same colour.

'21,

0. 7nvelop contains / cards labeled as L, L, F, F,

F "hereas envelop R contains 0 cards labeledas L, F, F, F, F, F. A card is picked at random

from envelop and envelop R. +y dra"ing a tree

diagram, f ind the probability that both of thecards are of different letters.

,31,

;. A fair dice is rolled three times. +y dra"ing a tree

diagram, find the probability that the number is

obtained only once.

2. The probability of obtaining a spoilt orange from a

basket is.

1. If three oranges are selected, find the

probability that only an orange is spoilt.

Probability 8


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;''/

0.';

7.( Past Years) Act#al SP" *#estions

Task : Ans"er all the Buestions belo".

1. SP" 2++( ,-o.2( & Paper 1

A box contains 0 "hite marbles and k black

marbles. If a marble is picked randomly from the box, the probability of getting a black marble is

/

,. ind the value of k . 4 marks

k $ 8

'. SP" 2++/ ,-o. 2( & Paper 1

The follo"ing table sho"s the number of coloured

cards in a box

0olo#r -#mber o! 0ards

+lack /

+lue

Fello"

T"o cards are dra"n at random from the box. ind

the probability that both cards are of the same

colour. 4 marks

0018

. SP" 2++ ,-o. 2% & Paper 1

The probability that (amid Bualifies for the final of a

track event is

/

' "hile the probability that ohan

Bualifies is,

1. ind the probability that

a! both of them Bualify for the final,

b! only one of them Bualifies for the final.

4 marks

. SP" 2++ ,-o. 2( & Paper 1

The probability of #arah being chosen as a

school prefect is

/

,

"hile the probability of

Aini being chosen is1'

;. ind the probability

that a! neither of them is chosen as a school

prefect,

b! only one of them is chosen as a school prefect. 4 marks

Probability 13


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a!1/'

b!1/;

a!01

b!03'8

7./ En'anced E$ercise 3it' *#estions o! SP" 4ormat

Task : Ans"er all the Buestions belo".

1. A bag contains 0 blue marbles and k red marbles. If

a marble is picked randomly from the bag, the

probability of picking a red marble is

/. ind the

value of k .

8

'. T"o dice, one "hite and one black, are rolled

together. -alculate the probability that the scoreon the "hite dice is t"ice the score on the blackdice.

1'1

. A box contains 3 marbles. #ome are green and

some are red. If a marble is dra"n at random from

the box, the probability that a green marble dra"n

is'

/. -alculate

a! the number of red marbles in the box,

b! the number of red marbles that have to be

added to the box such that the probability to

dra" a red marble becomes',1/ .

. +ag I contains ' blue marbles and 0 black marbles

"hile bag II contains blue marbles and black

marbles. If a marble is chosen at random from each

bag, calculate the probability that

a! both the marbles are black,

b! the marble from bag I is blue and the marble

from bag II is black.

c! At least one of the marbles chosen is black.

Probability 11


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a! ' b! 0 a! ;,

b! ;1

c! '2'/

/. T"o six?faced unbiased dice are rolled together.

-alculate the probability that

a! the sum of t"o numbers is 2.

b! The difference of t"o numbers is /,c! The sum of t"o numbers is 2 or The

difference of t"o numbers is /.

a! ,0/

b! 121 c! ,0

;

0. In a soccer match bet"een team A and team +, the

result can be a dra" or a "in for team A or a "in for

team +. The probability that team A and team + "ill

"in are1 1

and 'respectively. In t"o matches,

calculates the probability that team A "ill "in once

and dra" once.

121

;. A marble is dra"n at random from a box containing

black marbles, green marbles and / "hite

marbles.

a! Ghat is the probability of dra"ing a black or agreen marbleH

b! Ghat is the probability of dra"ing neither a

black nor a "hite marbleH

2. +ox - contains marbles of "hich are black and 0

are yello". T"o marbles are dra"n at random, one

after another "ith replacement. Determine the

probability thata! both the marbles are black.

b! the t"o marbles in different colour.

c! at least one of the marbles is yello".

Probability 1'


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a! '3,

b! '3,

a!,

1 b! 0

/

Probability 1

7 probability.doc

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