worksheet 9.1 probability distributions name: · 2019-10-13 · random variable. (c) not a random...
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Maths Quest Maths B Year 12 for Queensland Chapter 9 Probability Distributions WorkSHEET 9.1 1
WorkSHEET 9.1 Probability distributions Name: _________________________ 1 Which of the following are discrete random
variables? (a) The heights of students in a Year 12
class. (b) The weights, to the nearest kg, of
students in a Year 12 class. (c) The number of runs scored in a cricket
test match in Brisbane in 2002. (d) The number of consecutive heads
obtained when repeatedly tossing a coin. (e) The price, per litre of petrol, in randomly
selected stations in Queensland. (f) The actual volume of petrol in the
1000 litre ‘unleaded petrol’ tanks at those same stations after being filled by a tanker.
(a) Height is a continuous. (b) Weight, to the nearest kg, is a discrete
random variable. (c) Not a random variable, since match has
already occurred. (d) Although infinite, still a discrete random
variable. (e) Discrete, since price is always quoted to
the nearest $0.001 (f) Varies continuously, even when ‘full’ due
to continuous pressure and temperature variation.
2 You roll a die 3 times. What is the probability you get at least 2 sixes?
Using Yr 10 Probability Rules:
𝑃(> 2𝑠𝑖𝑥𝑒𝑠) = 𝑃(2𝑠𝑖𝑥𝑒𝑠) + 𝑃(3𝑠𝑖𝑥𝑒𝑠) = 𝑃(6,6,60) + 𝑃(6,60, 6) + 𝑃(60, 6,6) + 𝑃(6,6,6)
=16 ×
16 ×
56 +
16 ×
56 ×
16 +
56 ×
16 ×
16 +
16 ×
16 ×
16
=227
3 Validate that last answer using Binomial Probability.
Using Binomial:
𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙𝐶𝐷 =2, 3, 3,16>
= 0.074074074
=227
4 You are guessing the answers to a 3 question multiple choice test where each question has 5 possible answers. What is the probability you pass?
Using Yr 10 Probability Rules: 𝑃(> 2𝑐𝑜𝑟𝑟𝑒𝑐𝑡) = 𝑃(2𝑐𝑜𝑟𝑟𝑒𝑐𝑡) + 𝑃(3𝑐𝑜𝑟𝑟𝑒𝑐𝑡) = 𝑃(⇃, ⇃,×) + 𝑃(⇃,×, ⇃) + 𝑃(×, ⇃, ⇃) + 𝑃(⇃, ⇃, ⇃)
=15 ×
15 ×
45 +
15 ×
45 ×
15 +
45 ×
15 ×
15 +
15 ×
15 ×
15
=13125
Maths Quest Maths B Year 12 for Queensland Chapter 9 Probability Distributions WorkSHEET 9.1 2
5 Validate that last answer using Binomial Probability.
Using Binomial:
𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙𝐶𝐷 =2, 3, 3,15>
= 0.104
=13125
6 Consider the following probability table:
𝑥 0 1 2 3 𝑃(𝑋 = 𝑥) 0.1 0.2 0.3 𝑘
Determine the value of 𝑘.
Because;
H𝑃(𝑋 = 𝑥) = 1
0.1 + 0.2 + 0.3 + 𝑘 = 1
𝑘 = 0.4
7 Consider the following probability table:
𝑥 2 3 4 5 𝑃(𝑋 = 𝑥) 2𝑘 𝑘 3𝑘 4𝑘
Determine the value of 𝑘, hence determine the probability table values.
Because;
H𝑃(𝑋 = 𝑥) = 1
2𝑘 + 𝑘 + 3𝑘 + 4𝑘 = 1
𝑘 =110
𝑥 2 3 4 5 𝑃(𝑋 = 𝑥) 0.2 0.1 0.3 0.4
8 Consider the following probability table:
𝑥 0 1 2 3 𝑃(𝑋 = 𝑥) 0.1 0.2 0.3 0.4
Determine 𝐸(𝑥).
𝐸(𝑥) =H𝑥 × 𝑃(𝑋 = 𝑥)
0 × 0.1 + 1 × 0.2 + 2 × 0.3 + 3 × 0.4
∴ 𝐸(𝑥) = 2
Maths Quest Maths B Year 12 for Queensland Chapter 9 Probability Distributions WorkSHEET 9.1 3
9 Consider the following probability table:
𝑥 2 3 4 5 𝑃(𝑋 = 𝑥) 0.2 0.1 0.3 0.4
Determine 𝐸(𝑥).
𝐸(𝑥) =H𝑥 × 𝑃(𝑋 = 𝑥)
2 × 0.2 + 3 × 0.1 + 4 × 0.3 + 5 × 0.4
∴ 𝐸(𝑥) = 3.9
Maths Quest Maths B Year 12 for Queensland Chapter 9 Probability Distributions WorkSHEET 9.1 4
10 I throw a coin twice. How many Heads do I expect to get?
NO, you can’t just say you expect to get 1 heads! We need a Probability Table and then calculate the Expected value! As we are doing something 2 times, the values for the probability table can come from a year 8 two-way table!
H T
H HH HT
T TH TT
State the use of 𝑃 = LMNM
Hence we arrive at the Probability Table
𝑥 0 1 2 𝑃(𝑋 = 𝑥) 0.25 0.5 0.25
𝐸(𝑥) =H𝑥 × 𝑃(𝑋 = 𝑥)
0 × 0.25 + 1 × 0.5 + 2 × 0.25
∴ 𝐸(𝑥) = 1
Phew! We did know it was going to be 1J I love it when different mathematical techniques get the same correct answer!
Maths Quest Maths B Year 12 for Queensland Chapter 9 Probability Distributions WorkSHEET 9.1 5
11 I throw a coin thrice (three times). How many Heads do I expect to get?
Use the Binomial Probability Rule to get your values for the Probability Table;
O𝐶P =12>
P
=12>
O
=18
O𝐶R =12>
R
=12>
S
=38
O𝐶S =12>
S
=12>
R
=38
O𝐶O =12>
O
=12>
P
=18
Hence we arrive at the Probability Table
𝑥 0 1 2 3 𝑃(𝑋 = 𝑥) 1
8 38
38
18
𝐸(𝑥) =H𝑥 × 𝑃(𝑋 = 𝑥)
0 ×18 + 1 ×
38 + 2 ×
38 + 3 ×
18
∴ 𝐸(𝑥) = 1.5
Maths Quest Maths B Year 12 for Queensland Chapter 9 Probability Distributions WorkSHEET 9.1 6
12 Five coins are tossed simultaneously and the number of heads recorded. (a) Tabulate the probability distribution for
the number of heads.
(b) Draw a probability distribution graph of the outcomes.
(a) Pr(5 heads) = Pr(0 heads) = =
There are 5 ways to get 4 heads or 1 head. There are 10 ways to get 3 heads or 2 heads.
** Silly textbook … this is like throwing 1 coin 5 times, so the values in the table below come from using the Binomial Probability rule! **
(b)
521
321
321
325
3210
3210
325
321)Pr(
543210
x
x
3213223233243253263273283293210
X= 0 1 2 3 4 5
Maths Quest Maths B Year 12 for Queensland Chapter 9 Probability Distributions WorkSHEET 9.1 7
13 Calculate the Variance of the following Probability Table.
𝑥 0 1 2 3 𝑃(𝑋 = 𝑥) 0.1 0.2 0.3 0.4
Using the rule on the Formula Sheet;
𝑉𝑎𝑟(𝑋) =H𝑝(𝑥 − 𝜇)S Need 𝜇:
𝐸(𝑥) =H𝑥 × 𝑃(𝑋 = 𝑥)
= 0 × 0.1 + 1 × 0.2 + 2 × 0.3 + 3 × 0.4
𝐸(𝑥) = 2
∴ 𝜇 = 2 𝑉𝑎𝑟(𝑋) = 0.1(0 − 2)S + 0.2(1 − 2)S
+ 0.3(2 − 2)S + 0.4(3 − 2)S
∴ 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 1
14 Calculate the Variance of the following Probability Table.
𝑥 0 1 2 3 𝑃(𝑋 = 𝑥) 0.1 0.2 0.3 0.4
Using other rule. 𝑉𝑎𝑟(𝑋) = 𝐸(𝑋S) − X𝐸(𝑋)YS
𝐸(𝑋) =H𝑥 × 𝑃(𝑋 = 𝑥)
= 0 × 0.1 + 1 × 0.2 + 2 × 0.3 + 3 × 0.4
𝐸(𝑥) = 2
𝐸(𝑋S) =H𝑥S × 𝑃(𝑋 = 𝑥)
= 0S × 0.1 + 1S × 0.2 + 2S × 0.3 + 3S × 0.4
𝐸(𝑥) = 5
𝑉𝑎𝑟(𝑋) = 5 − 2S
∴ 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 1
Maths Quest Maths B Year 12 for Queensland Chapter 9 Probability Distributions WorkSHEET 9.1 8
15 Calculate the Variance of the following Probability Table.
𝑥 0 1 2 3 𝑃(𝑋 = 𝑥) 0.1 0.2 0.3 0.4
Using the Casio … input List 1 and List 2 ensure frequency is set to List 2 calculate 1-VAR
𝐸(𝑋) = 2 𝐸(𝑋S) = 5
Use rule: 𝑉𝑎𝑟(𝑋) = 𝐸(𝑋S) − X𝐸(𝑋)YS
𝑉𝑎𝑟(𝑋) = 5 − 2S
∴ 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 1
16 Calculate the Standard Deviation of the following Probability Table.
𝑥 0 1 2 3 𝑃(𝑋 = 𝑥) 0.1 0.2 0.3 0.4
𝑆𝐷 = [𝑉𝑎𝑟(𝑋) Refer to previous calculations for Variance, hence
𝑆𝐷 = √1 = 1
17 Calculate the Standard Deviation of the following Probability Table.
𝑥 0 1 2 3 𝑃(𝑋 = 𝑥) 0.1 0.2 0.3 0.4
Using the Casio … input List 1 and List 2 ensure frequency is set to List 2 calculate 1-VAR
𝜎 = 1
18 Any chance you get, when you see a Probability Table, calculate the variance and SD in different ways. You need lots of practice!
Maths Quest Maths B Year 12 for Queensland Chapter 9 Probability Distributions WorkSHEET 9.1 9
19 A die is ‘fixed’ so that certain numbers will appear more often. The probability that a 6 appears is twice the probability of a 5 and 3 times the probability of a 4. The probabilities of 3, 2 and 1 are unchanged from a normal die. The probability distribution table is given below.
Find: (a) The value of x in the probability
distribution and hence complete the probability distribution.
(b) The probability of getting a ‘double’ with two of these dice. Compare with the ‘normal’ probability of getting a double.
(a) Since the sum of the probabilities must be 1,
+ + + + + x = 1
Putting over a common denominator,
= 1
Collect like terms and remove fraction, 3 + 11x = 6
x =
Note Pr(5) = , Pr(4) = =
(b)
(c) Probability of a ‘double’ is given by
Pr(1) ´ Pr(1) + Pr(2) ´ Pr(2) + … +Pr(6) ´ Pr(6)
Pr(double) = ´ + ´ + ´ +
´ + ´ + ´
Convert each product to a decimal.
Pr(double) = 0.02777 + 0.02777 + 0.02777 + 0.00826 + 0.01860 + 0.07438
Pr(double) = 0.1846 The ‘normal’ probability of a double is
0.1666.
xxxx
x
2361
61
61)Pr(
654321
61
61
61
3x
2x
6632111 xxx +++++
113
223
333
111
11
3
22
3
11
1
6
1
6
1
6
1)Pr(
654321
x
x
61
61
61
61
61
61
111
111
223
223
113
113
Maths Quest Maths B Year 12 for Queensland Chapter 9 Probability Distributions WorkSHEET 9.1 10
20 Show that p(x) = , for x = 1,2, … 6
is a probability distribution. State Pr (2 < x < 6)
Set up a table of probabilities
Calculate the sum of the probabilities.
Sum = = 1
Because ∑𝑃(𝑋 = 𝑥) = 1, we can confirm that
the given 𝑝(𝑥) IS a probability distribution.
Obtaining data from the table;
Pr (2 < x < 6) = + +
=
21 Find the expected value of the following discrete probability distribution.
Use formula E(X) = SxP(X = x) E(X) = 1(0.1) + 2(0.15) + 3(0.25)
+ 4(0.05) + 5(0.45) E(X) = 0.1 + 0.3 + 0.75 + 0.2 + 2.25 = 3.6
6634 -x
6621
6617
6613
669
665
661)Pr(
654321
x
x
66211713951 +++++
669
6613
6617
6639
45.005.025.015.01.0)Pr(54321
xXx=
Maths Quest Maths B Year 12 for Queensland Chapter 9 Probability Distributions WorkSHEET 9.1 11
22 Consider the following gambling game, based on the outcome of the total of 2 dice:
– if the total is a perfect square, you win $4 – if the total is 2, 6, 8 or 10, you win $1 – otherwise, you lose $2.
(a) Find the expected value of this game. (b) Determine if it is a fair game.
(a) Set up the probability distribution table.
Add a row, which indicates win (+) or loss(–).
Use formula E(gain) = SGain(x) Pr(X = x)
E(gain) = 1( ) –2( ) + 4( ) –2( )
+ 1( ) –2( ) +1( ) + 4( )
+ 1( ) –2( ) –2( )
E(Gain) =
E(Gain) = – =
E(Gain) = 0.333 (b) This game is ‘unfair’ — you stand to gain
about $0.33 every time you play!
23 Find the missing profit (or loss) so that the following probability table has an expected value of 0.
Let y = profit/loss for x = 10.
E(Gain) = SGain(x) Pr(X = x) E(Gain) = –3(0.1) + 4(0.06) – 2(0.25) + 5(0.16) –
8(0.09) + 12(0.21) + y(0.13) Simplify and set E(gain) = 0 –0.3 + 0.24 – 0.5 + 0.8 – 0.72 + 2.52 + 0.13y = 0 2.04 + 0.13y = 0 Solve for y
361
362
363
364
365
366
365
364
363
362
361)Pr(
12111098765432xx
22141212421Gain361
362
363
364
365
366
365
364
363
362
361)Pr(
12111098765432
-----
xx
361
362
363
364
365
366
365
364
363
362
361
3631655121 +++++
36241284 ++++
-
3642
3630
3612
1285243Gain13.021.009.016.025.006.01.0)Pr(10987654
---= xXx
69.1513.004.2
-=-
=y
Maths Quest Maths B Year 12 for Queensland Chapter 9 Probability Distributions WorkSHEET 9.1 12
24 For the following probability distribution calculate: (a) E(X) (b) E(2X)
(c) E(X + 2) (d) E(X2)
(e) E(X2) – [E(X)]2.
Using a table of values (or the Maths Quest spreadsheet ‘Prob distribution’): (a) E(X) = 1.22
(b) E(2X) = 2.44
(c) E(X + 2) = 3.22
(d) E(X2) = 5.24
(e) E(X2) – [E(X)]2 = 3.7516
07.07.11.2.17.12.21.05.)Pr(54321012
xXx=
--
Maths Quest Maths B Year 12 for Queensland Chapter 9 Probability Distributions WorkSHEET 9.1 13
25 Three players play the following game for a prize pool of $210. Alice tosses a coin — if it is heads she wins. If not, then Betty tosses the coin — if it is heads she wins. If not, then Carla tosses the coin — if it is heads she wins. If not, then Alice tosses the coin again, winning if it is a head … and so on. Find the expected value of each person in this game.
Because, in theory, this game could go on forever, determine (relative) probabilities as follows. In round 1,
Alice has a 12 chance of winning, while Betty has a
12 ´ 12 chance, and Carla has a 12 ´ 12 ´ 12 chance.
In Round 2,
Alice has a 12 ´ 12 ´ 12 ´ 12 chance … and so on.
These probabilities are tabulated below.
By looking at each row, the probabilities are in the ratio of 4 : 2 : 1 Thus Alice has 4 ‘chances’, Betty has 2 and Carla has 1.
E(Alice) = 47 (210) = $120
E(Betty) = 27 (210) = $60
E(Carla) = 17 (210) = $30
40961
20481
102414
5121
2561
12813
641
321
1612
81
41
211
CarlaBettyAliceRound