STAT 1430 Binomial Distribution

Unless otherwise noted you may use either the formula or table to find binomial probabilities
where appropriate. In either case show your work. If you use a number off the table, state this.
Have your formula sheet and binomial table out and ready to use for these problems.

NOTE: THE BINOMIAL TABLE YOU SHOULD USE IS ON CARMEN UNDER COURSE
INFORMATION.

1. Suppose 30% of OSU students watch reality TV shows of some kind every week. Three OSU
students are selected at random and asked if they watch reality TV of some kind each week. Let
X be the number of students in the sample who answer YES.

Find the probability distribution for X using two methods: 1) use the formulas and work it
out; AND 2) use the binomial table and calculate from there. (Remember what a probability
distribution means – see notes from discrete random variables.)

USING FORMULAS:

USING TABLE: (Remember you need to do some subtraction to get the probabilities you
need.)

2. Suppose 20% of all football fans get to attend a game in person. You sample 100 fans at random
and want to find the probability that at least 15% of them get to attend a game in person. This
problem represents a binomial distribution with n = 100. Which of these numbers represents the
value of “p”?
0.15 0.20

3. Explain why X DOES NOT have a binomial distribution in each of the situations below. (Hint,
check the criteria; if you find one that isn’t met, X isn’t binomial.)

a. Suppose a police officer takes a two-hour time period and records the number of
vehicles traveling on US 131 that exceed the speed limit (where the speed limit is 70
miles per hour). Let X denote the number of vehicles that were exceeding the limit.

b. Suppose 10% of OSU business students major in international marketing. You keep
sampling students at random from our class until you find someone who is majoring in
international marketing. Let X = the number of students you have to sample.

c. You have 10 people working for you, 5 men and 5 women, and you have to choose
different people be on a committee that no one wants to be on. You put their names into
a hat and pull out 3 names, one by one. Let X = number of women who end up on the
committee.
STAT 1430 Binomial Distribution

d. You take a random sample of 10 college students from stat 1430 and record their status
(freshman, sophomore, etc.).

e. Suppose a police officer takes a two-hour time period and records the number of
vehicles traveling on US 131 that exceed the speed limit (where the speed limit is 70
miles per hour). Let X denote the number of vehicles that were exceeding the limit.
Explain why X does NOT have a binomial distribution.

4. Which of the following is not a characteristic of a binomial distribution?

a. There is a set of n trials
b. Each trial results in more than one possible outcome.
c. The trials are independent of each other.
d. Probability of success p is the same from one trial to another.

5. Which of the following has a Binomial distribution?

a. The number of customers arriving at a gas station on July 4
b. The number of people against a smoking ban out of a random sample of 100.
c. The number of telephone calls received by a switchboard in a specified time period
d. All of the above have a binomial distribution.

6. The standard deviation of a binomial probability distribution with n trials and probability p of
success is:
a. np
b. square root of (np(1-p))
c. n + p
d. np(1-p)

7. One out of four of the students in an English class is an international student. Take a random
sample of 100 students from this class and let X = the number of international students. The mean
of X is what? (Note! You are not given p directly but you can find it):
a. 18.75
b. 4.33
c. 25
d. 5

8. If 30% of Americans own a pet and you select 100 Americans at random and let X = number who
own a pet. What is the mean and standard deviation of X?

For the following problems show your work, and make sure you use probability notation in your
work. For example, if trying to find the probability that X is greater than 2 write: P(X>2).

9. 30% of OSU students have cars that are black. Suppose you randomly sample 5 OSU students, and
let X be the number of students in the sample with black cars. Find the probabilities of the
following events.
a. Exactly 3 students in the sample have black cars.
b. More than 3 students in the sample have black cars.
c. Less than half of the students in the sample have black cars.
STAT 1430 Binomial Distribution
d. At least one student in the sample has a black car.

10. There are two conditions that you have to check before using the normal approximation when X is
binomial.
a. Write down the two conditions.
b. Why do we have to check BOTH conditions rather than just one, to be able to use the
normal approximation? (Including an example with your explanation is OK.)

11. Suppose you flip a fair coin 7 times, and let X be the total number of tails. What is the probability
that X is greater than 1? (Hint, what is p when you flip a fair coin?)

12. Suppose it is reported that 10% of OSU business students major in international marketing. You
sample 200 OSU business students at random. Let X = the number of students in the sample who
major in international marketing.
a. How many students in your sample do you EXPECT to major in int’l marketing, according
to the above information?
b. What is the probability that your sample finds that at least 15 students major in international
marketing?
c. Are your findings from part (b) typical of what you EXPECTED to get in part (a)? Why or
why not?

13. Suppose X is binomial with p = .90. What does n have to be (at a minimum) to use the normal
approximation for X?

14. Suppose X is binomial with p = .10. What does n have to be (at a minimum) to use the normal
approximation for X?

15. Explain why the same n works for both of the previous problems.

16. Suppose 20% of a bank’s customers report being unsatisfied. You take a sample of 20 customers
at random and count the number of unsatisfied customers.
a. Is the distribution of the number of unsatisfied customers from this sample binomial?
Justify.
b. What is the probability that 5 customers are unsatisfied?
c. USING THE FORMULA AND THE TABLE TO VERIFY: What is the probability that at
most 2 customers are unsatisfied? Show your work.

17. Suppose a media report claims that 25% of children are addicted to video games. You think the
percentage is higher than that. You take a random sample of 200 children and find that at least 70
of them are addicted to video games.
a. What is the chance of this result happening?
b. Based on your results do you believe the claim that 25% of children are addicted to video
games? Why or why not?

18. Suppose we have a 20 question MCTF test where each question has 4 possible answers.
a. If you guess on all the questions, and you let X be the number you get right, what is the
mean of X? Why does this number make sense?
STAT 1430 Binomial Distribution
b. If you guess on all the questions, and you let X bet the number you get right, what is the
standard deviation of X?
c. If someone got 10 questions right on this test, do you think they were guessing? Why or
why not? Use the binomial table to help you decide.

19. The national proportion of adults who are concerned about nutrition is 0.35. You take a random
sample of 5 adults and count the number who are concerned about nutrition (call this number X).
a. What is µx?
b. What is σx?
c. What is P(X = 3)?
d. What is the probability that X is at most 3? (Use the table)
e. What’s the probability that X is at least 3? (Use the table)
f. Explain why the previous two problems are NOT complements of each other.

20. Suppose 30% of OSU students watch reality TV shows of some kind every week. Ten OSU
students are selected at random and asked if they watch reality TV of some kind each week. You
want the probability that at least 4 of them say yes.
a. What does X represent in this problem?
b. Show BRIEFLY that X has a binomial distribution.
c. Find the probability that at least 4 of the students say yes.
d. Could a person use the normal approximation to answer this question? Explain briefly.

21. Again suppose 30% of OSU students watch reality TV shows of some kind every week. Now 100
OSU students are selected at random and asked if they watch reality TV of some kind each week.
X = number in the sample who watch reality TV of some kind each week.
a. Find the mean and SD of X.
b. Justify that X has an approximate normal distribution.
c. Use the normal approximation to find the probability that at least 40 of the students in the
sample say yes.
d. Explain why being able to use the normal approximation for this question is a GOOD
THING. (What would you have to do otherwise?)