HW 5

STAT 3021

Q1. For each of these following examples, define the random variable of interest and recognize

their distributions respectively (one of the distributions learnt in class- Binomial, Multinomial,

Hypergeometric, Negative Binomial, Geometric, Poisson). Do not solve the problems in this part.

For example, you find a r.v. X given in the question follows a binomial distribution. Then you

need to specify the parameters B(n =?, p =?) according to our slides.

1. An oil company conducts a geological study that indicates that an exploratory oil well should

have a 20% chance of striking oil. What is the probability that the first strike comes on the

third well drilled?

2. Suppose you are given a 3 question multiple-choice test. Each question has 4 responses and

only one is correct. Suppose you want to find the probability that you can just guess at the

answers and get 2 questions right.

3. A crate contains 50 light bulbs of which 5 are defective and 45 are not. A Quality Control

Inspector randomly samples 4 bulbs without replacement. What is the probability that 4

bulbs are defective?

4. Let’s say that 80% of all business startups in the IT industry report that they generate a

profit in their first year. If a sample of 10 new IT business startups is selected, find the

probability that exactly seven will generate a profit in their first year.

5. Bob is a high school basketball player. He is a 70% free throw shooter. That means his

probability of making a free throw is 0.70. During the season, what is the probability that

Bob makes his third free throw on his fifth shot?

6. Your basketball team is playing a series of 5 games against your opponent. The winner is

those who wins more games (out of 5). Let assume that your team is much more skilled and

has 75% chances of winning. It means there is a 25% chance of losing. What is the probability

of your team gets 3 wins?

7. Births in a hospital occur randomly at an average rate of 1.8 births per hour. What is the

probability of observing 4 births in a given hour at the hospital?

8. Bob is a high school basketball player. He is a 70% free throw shooter. That means his

probability of making a free throw is 0.70. During the season, what is the probability that

Bob makes his first free throw on his fifth shot?

9. A box of candies has many different colors in it. There is a 15% chance of getting a pink

candy. What is the probability that exactly 4 candies in a box are pink out of 10?

10. A deck of cards contains 20 cards: 6 red cards and 14 black cards. 5 cards are drawn randomly

without replacement. What is the probability that exactly 4 red cards are drawn?

1

STAT 3021

HW 5

11. In a certain town, 40% of the eligible voters prefer candidate A, 10% prefer candidate B, and

the remaining 50% have no preference. You randomly sample 10 eligible voters. What is the

probability that 4 will prefer candidate A, 1 will prefer candidate B, and the remaining 5 will

have no preference?

12. A small voting district has 101 female voters and 95 male voters. A random sample of 10

voters is drawn. What is the probability exactly 7 of the voters will be female?

13. Suppose you knew that the mean number of calls to a fire station on a weekday is 8. What

is the probability that on a given weekday there would be 11 calls?

14. An experiment of drawing a random card from an ordinary playing cards deck is done with

replacing it back. This was done ten times. Find the probability of getting 2 spades, 3

diamond, 3 club and 2 hearts.

15. It is known that 20% of products on a production line are defective. Products are inspected

until first defective is encountered. What is the probability that the first defective was found

after inspecting exactly 3 products?

16. A man was able to complete 3 files a day on an average. Find the probability that he can

complete 5 files the next day.

Q2. Answer the following questions from the book:

• Section 5.2(Binomial and multinomial): 5.4, 5.18, 5.79

• Section 5.3(Hypergeometric): 5.32, 5.40

• Section 5.4(Negative binomial and geometric): 5.90,5.91

• Section 5.5(Poisson): 5.58, 5.62, 5.68

2

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