Practice Questions
1. The HR department of a company has two men and three women. The manager of the
department plans to assign a new project to two employees.
(a) What is the probability that the two employees are a man and a woman?
2. In a lottery conducted to benefit a local charity, 600 tickets are to be sold at $10 each.
There is only one winner, and the prize is $2400.
(a) What could be the lowest price of the ticket?
(b) Suppose John purchases two tickets, what is his expected gain?
3. ABC university has 1000 students. Five hundred of the students took M0517 last semester.
You are interested in the average score of students taking the course. You randomly asked
200 students from that university. Only 100 of the 200 students took M0517 last semester,
and they told you their grades.
(a) What is the population in the scenario?
(b) What is the sample in the scenario?
4. Suppose events A and B are not empty sets. If events A and B are independent,
(a) will the two events mutually exclusive? Justify your answer.
5. Mary creates a gambling game that requires two flips of a fair coin. If the first toss of a
coin is a head, you win 10 dollars. If it is a tail, you lose 6 dollars. On the second toss, a
head will pay 15 dollars and a tail will cost you 12 dollars. Suppose John plays the game.
(a) How much money is John expected to earn or lose if his first toss shows a head?
6. Suppose a fair die has 7 sides, which are inscribed English alphabets:
A,A,B,B,C,C,D
Let X be a random variable representing the alphabet order plus two.
(a) Find E (X).
(b) Find V ar (X).
7. Suppose a population X has the following values : {7, 2, 6, 2, 3}.
(a) What is the median of X?
(b) What is the variance of X?
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8. Suppose A and B are two events. If P (A) = 0.4, P (B|A) = 0.35 and P (A ∪ B) =
0.69,
(a) find P (A|B).
(b) find P (A ∩ B|A ∪ B).
9. In each of the following cases, indicate whether classical, empirical, or subjective probability is used:
(a) A baseball player gets a hit in 30 out of 100 times at bat. Therefore, the probability
he gets a hit in his next at bat is 0.3.
(b) A seven-member committee of students is formed to study environmental issues.
What is the likelihood that any one of the seven is randomly chosen as the spokesperson?
10. Suppose A and B are two events, and S is the sample space. A = {1, 2, 3, 4, 5}. B =
{4, 5, 6, 7, 8}. S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
(a) Find p (A ∪ B).
(b) Are A and B collectively exhausted? Justify your answer.
11. A department store reported that 30% of their sales are cash, 30% are paid with a credit
card, and 40% with a debit card. Twenty percent of the cash purchases, 90% of the
credit card purchases, and 60% of the debit card purchases are for more than $50. Steven
bought a new hat that cost $20.
(a) What is the probability that Steven paid in cash?
12. DJ LeMahieu of the New York Yankees had the highest batting average in the 2016 Major
League Baseball season. His average was .348. So assume the probability of getting a
hit is .348 for each time he batted. In a particular game, assume he batted three times.
Suppose we assume that each at bat is independent.
(a) What is the probability of getting at least one hit?
(b) Suppose he didn’t get any hit in his first and second at bat. What is the probability
of getting a hit in his third at bat?
13. Suppose that a random variable X has a discrete distribution with the following pmf:
f (x) =

cx2
for x = 0, 1, 2, 3
0
otherwise
(a) Find c.
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(b) Find E (X).
(c) Find V ar (X).
14. Suppose X = {1, 2, 3} and pX (1) = pX (2) = pX (3) = 1/3.
(a) Find E (2X + 1).
(b) Find V ar (2X + 1).
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