Proability Distribution Review -...

Name: ________________________ Class: ___________________ Date: __________ ID: A

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Proability Distribution Review

Problem

1. A sailing club has eight 4.6-m boats, eleven 5.0-m boats, five 5.2-m boats, and five 6.1-m boats. These boats are assigned randomly to members who want to go sailing on any given day.a) Make a table and a graph of the probability distribution for the length of an assigned boat.b) What is the expected length of an assigned boat?

2. A spinner has three equally-sized sectors, numbered 1 through 3.a) What is the probability that the arrow on the spinner will stop on an even number?b) What is the expected outcome?

3. To finish a board game, a player often has to roll a number with a pair of dice such that the player’s counter lands exactly on the last square of the board. Suppose a player is three squares from the end of the board. Calculate the probability distribution for each possible waiting time up to the player taking ten rolls to finish the game.

4. Suppose that one third of the cards in a scratch-and-win promotion gives a prize.a) What is the probability that you will not win a prize until your second try?b) What is the probability of winning within your first two tries?c) What is the expected number of cards you would have to try before winning a prize?

5. A 12-member jury for a criminal case will be selected from a pool of 15 men and 15 women.a) What is the probability that the jury will have 6 men and 6 women?b) What is the probability that at least 3 jurors will be women?c) What is the expected number of women?

6. Suppose that Bayanisthol, a new drug, is effective for 65% of the participants in clinical trials. If a group of fifteen patients take this new drug,a) what is the expected number of patients for whom the drug will be effective?b) what is the probability that the drug will be effective for less than half of them?

ID: A

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Proability Distribution ReviewAnswer Section

PROBLEM

1. ANS: a)

Length, x (m) Probability, P(x)4.6 8

295.0 11

295.2 5

296.1 5

29

b) E(X) = 4.68

29

Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜

+ 5.011

29

Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜

+ 5.25

29

Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜

+ 6.15

29

Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜

= 5.11 The expected length is 5.11 m.

REF: Applications OBJ: Section 7.1

ID: A

2

2. ANS:

a) The only even number on the spinner is 2, so the probability is 1

3.

b) E(X) = 11

3

Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜

+ 21

3

Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜

+ 31

3

Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜

= 2

REF: Applications OBJ: Section 7.1 3. ANS:

The probability of rolling a total of 3 with a pair of standard dice is 2

36, so p =

1

18 and q =

17

18. Using the

formula P(x) = qxp gives the following probabilities.

Waiting time, x Probability, P(x)0 0.055 55…1 0.052 46…2 0.049 55…3 0.046 80…4 0.044 20…5 0.041 74…6 0.039 42…7 0.037 23…8 0.035 16…9 0.033 21…


a) P(X = 1) =2

3×

1

3

=2

9or about 0.2222

b) P(X ≤ 1) =2

3

Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜

01

3+

2

3

Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜

11

3

=5

9or about 0.5556

c) E(X) =

2

3

1

3

= 2


ID: A

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5. ANS: Let the random variable X be the number of women on the jury.

a) P(X = 6) = 15C6 × 15C6

30C12

= 0.2896b) P(X ≥ 3) = 1− P(0) + P(1) + P(2)È

ÎÍÍÍ

˘˚˙̇˙

= 1− 15C0 × 15C12

30C12

− 15C1 × 15C11

30C12

− 15C2 × 15C10

30C12

= 0.9961

c) E(X) =15× 12

30= 6


a) E(X) = np= 15× 0.65= 9.75

b) E(X ≤ 7) = 15C0(0.65)0(0.35)15 + 15C1(0.65)1(0.35)14 + 15C2(0.65)2(0.35)13 + 15C3(0.65)3(0.35)12

+ 15C4(0.65)4(0.35)11 + 15C5(0.65)5(0.35)10 + 15C6(0.65)6(0.35)9 + 15C7(0.65)7(0.35)8

= 0.1132

This

probability can also be calculated using the binomcdf( function on a graphing calculator, the BINOMDIST function in a spreadsheet, or the binomialProbability function in Fathom™.



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Problem





