Quantitative Methods – I

Probability Theory: A few Practice Problems

Prob. 1

The probability that two products A and B, newly introduced by a company will be successful are 0.75 and 0.80 respectively. The probability that both will be successful is 0.7. What is the probability that

a) A alone will be successful?

b) B alone will be successful?

c) Neither one will be successful?

d) At least one will be successful?

Prob. 2

If A and B are independent events, show that A’ and B’ are also independent events.

Prob. 3

If P(AB) = 5/6, P(AB) = 1/3, and P(B’) = ½, then which of the following statements is true.

a) A and B are mutually exclusive.

b) P(A) = P(B)

c) A and B are independent.

d) B is a sub-event of A.

Prob. 4

A coin is tossed three times. If A is the event that Head occurs on each of the first two tosses, B is the event that a tail occurs on the third toss, and C is the event that exactly two tails occur in three tosses, show that a) events A and B are independent. B) events B and C are not independent.

Prob. 5

Three machines, A, B and C produce 50%, 30% and 20% of the total number of items of a factory. The percentage of defective items produced by these machines are: 3%, 4% and 5%, respectively. An item is chosen randomly.

a) Find the probability that the item is defective.

b) Suppose the item is found to be defective. What is the probability that item was produced by machine A? by machine B?

Prob. 6

The probabilities of three men hitting a target are 1/6, ¼ and 1/3, respectively. Each shoots once at the target.

a) Find the probability that exactly one person hits the target.

b) If only one person hits the target, what is the probability that it was the first person.

Prob. 7

Suppose you are eating at a pizza parlor with two friends. You have agreed to the following rule to decide who will pay the bill. Each person will toss a coin. The person who gets a result that is different from the other two will pay the bill. If all three tosses yield the same result, the bill will be shared by all. Find the probability that:

a) Only you will have to pay.

b) All three will share the bill.

Prob. 8

A white and a colored die are tossed.

a) If both dice are fair, assign probabilities to each elementary outcome.

b) Identify events A = {sum = 6}, B = (sum = 7}, C = {sum is even}, D = {same number on each die}

c) Obtain probabilities of above events.

Prob. 9

Explain why there must be a mistake in each of the following statements.

a) A compute repair person claims that the probability is 0.8 that the hard disk is working properly, 0.7 that the RAM is working properly, and 0.3 that both are working properly.

b) An accountant claims that the probability is 0.95 that there are no significant errors in her analysis, and 0.08 that there are one or two significant errors.

c) The marketing manager of a company claims that the probability is 0.7 that product A will be successful, 0.3 that product B will be successful, and 0.4 that both will be successful.

d) A real estate investor claims that the probability is 0.6 that a certain property can be sold for a profit within one year, and 0.3 that it cannot.

Prob. 10

The following table shows the probabilities concerning two events A and B.

BB’

A0.250.12??

A’??????

------

0.40 ??

a) Determine the missing probabilities in the table.

b) What is the probability that A occurs and B does not occur?

c) Find the probability that either A or B occurs.

d) Find the probability that one of these events occurs an the other does not.

Prob. 11

Of the 19 nursing homes in the city, 6 are in violation of sanitary standards, 8 are in violation of security standards, and 5 are in violation of both. If a nursing home is chosen at random, what is the probability that it is compliance with both security and sanitary standards?

Prob. 12

The Wimbledon men’s tennis championship ends when one player wins three sets.

a) How many elementary outcomes end in three sets? In four sets? In five sets?

b) If the payers are evenly matched, what is the probability that the tennis match ends in four sets.

Prob. 13

A box has 9 black balls and 6 red ones. You randomly select 4 balls from the box. What is the probability that

a) None of the balls is a red one.

b) At least two balls are red.

c) Assuming that the balls are selected sequentially, the first two balls selected are black and the next two are red.