# Chapter 5: Discrete Probability Distributions

1.  A random variable x has a binomial distribution with n=4 and p=1/6. What is the probability that x is 1?

1. 0.3458
2. 0.4158
3. 0.4358
4. 0.3858 X

Solution:

> dbinom(1, 4, 1/6)

[1] 0.3858025

2.  A random variable x has a binomial distribution with n=64 and p=0.65. What is the probability that x is 47 or less?

1. 0.9417 X
2. 0.9717
3. 0.8817
4. 0.9017

Solution:

> pbinom(47, 64, 0.65)

[1] 0.9416856

or

> sum(dbinom(0 : 47, 64, 0.65))

[1] 0.9416856

3.  A random variable x has a binomial distribution with n=100 and p=0.35.  What is the probability x falls in the range from 26 to 34, inclusive?

1. 0.3813
2. 0.5413
3. 0.4413 X
4. 0.4913

Solution:

> sum(dbinom(26 : 34, 100, 0.35))

[1] 0.4412901

or

> pbinom(34, 100, 0.35) - pbinom(25, 100, 0.35)

[1] 0.4412901

4.  A random variable x has a binomial distribution with n =28 and p=0.55. What is the probability that x will be greater than 18?

1. 0.1787
2. 0.1187 X
3. 0.2256
4. 0.0887

Solution:

> 1 - pbinom(18, 28, 0.55)

[1] 0.1187211

or

> sum(dbinom(19 : 28, 28, 0.55))

[1] 0.1187211

5.  Suppose x is a Poisson-distributed random variable with an expected value of 5 occurrences per interval. What is p(x=3)?

1. 0.2004
2. 0.1404 X
3. 0.1704
4. 0.0904

Solution:

> dpois(3, 5)

[1] 0.1403739

6.  Suppose x is a Poisson-distributed random variable with an expected value of 12 occurrences per interval. What is p(x<10)?

1. 0.2424 X
2. 0.2124
3. 0.2824
4. 0.2624

Solution:

> ppois(9, 12)

[1] 0.2423922

or

> sum(dpois(0 : 9, 12))

[1] 0.2423922

7.  Suppose x is a Poisson-distributed random variable with an expected value of 55 occurrences per interval. What is p(45<x<60)?

1. 0.6055
2. 0.6755
3. 0.6355 X
4. 0.6955

Solution:

> ppois(59, 55) - ppois(45, 55)

[1] 0.6354798

or

> sum(dpois(46 : 59, 55))

[1] 0.6354798

8.  Suppose x is a Poisson-distributed random variable with an expected value of 105 occurrences per interval. What is p(x>90)?

1. 0.9641
2. 0.8741
3. 0.8341
4. 0.9241 X

Solution:

> ppois(90, 105, lower.tail =  FALSE)

[1] 0.9240666

or

> 1 - ppois(90, 105)

[1] 0.9240666

9.  An urn has 10 marbles: 3 red, 7 black.  If we draw a random sample of 4, what is the probability we will end up with 2 red and 2 black?

1. 0.1787
2. 0.1187 X
3. 0.2256
4. 0.0887

Solution:

> dhyper(2, 3, 7, 4)

[1] 0.3

or

> choose(3, 2) * choose(7, 2) / choose(10, 4)

[1] 0.3

10.   Suppose 10 cards are drawn from a deck of 52 cards consisting of 13 hearts, 13 diamonds, 13 clubs, and 13 spades. What is the probability that the hand of 10 cards will include 3 hearts, 3 diamonds, 2 clubs, and 2 spades?

1. 0.0315 X
2. 0.0515
3. 0.0815
4. 0.0215

Solution:

> choose(13, 3) * choose(13, 3) * choose(13, 2) * choose(13, 2) / choose(52, 10)

[1] 0.03145677