Decision Making and Probability

1. The probability of an event
a. is the sum of the probabilities of the sample points in the event.
b. is the product of the probabilities of the sample points in the event.
c. is the maximum of the probabilities of the sample points in the event.
d. is the minimum of the probabilities of the sample points in the event.

2. If A and B are independent events.
a. A and B are mutually exclusive.
b. A and B may or may not be mutually exclusive
c. A and B cannot be mutually exclusive events.
d. either P(A) = 0 or P(B) = 0.

3 If P(A|B) = .4, then
a. P(B|A) = .6
b. P(A)*P(B) = .4
c. P(A) / P(B) = .4
d. None of the alternatives is correct.

4 If P(A) = 0.38, P(B) = 0.83, and P(A  B) = 0.27; then P(A  B) =
a. 1.21
b. 0.94
c. 0.72
d. 1.48

5 The statement "If P(AB) = P(A) + P(B), then A and B are mutually exclusive" is:

a. always true
b. always false
c. only true when A and B are independent events
d. sometimes true, depending on the value of P(A|B)

6. You are taking two courses during winter session, math and history, and your subjective assessment of your performance is

Event Probability
fail both courses .05
fail math (irrespective of whether or not you fail history as well) .15
fail history (irrespective of whether or not you fail math as well) .08

a) What is the probability of failing math only (that is, you fail math but pass history)?

b) What is the probability of passing either course?

(Note: this is a difficult question, and it's strongly suggested that you draw a Venn diagram and / or create a joint probability table with failing/passing math in one dimension and failing/passing history in the other)

7. Through a telephone survey, a low-interest bank credit card is offered to 400 households. The responses are as tabled.

Income  $40,000 Income > $40,000
Accept offer 40 30
Reject offer 210 120

a) If someone who accepted the offer is selected at random, what is the probability that he earns at least $40,000?

b) What is the probability that one of the respondents, selected at random, accepted the offer?

8. It is estimated that 3% of the athletes competing in a large tournament are users of an illegal drug to enhance performance. The test for this drug is 90% accurate that is, the test will correctly identify a drug user as testing positive 90% of the time and will also correctly identify a non drug user as testing negative 90% of the time (that is:

P(tests positive | user) = 0.9 P(tests negative | non-user) = 0.9

Compute the probability that an athlete who tests positive is actually a user; that is, P(user | tests positive)

9. A video rental store has three video cameras available for customers to rent. Historically, demand for cameras during the June 'wedding season' has followed the distribution in the table below. The revenue per rental is $40. If a customer wants a camera and none is available, the store gives a $15 coupon for tape rental.
Demand Relative Frequency Revenue Cost
0 .25 0 0
1 .30 40 0
2 .20 80 0
3 .10 120 0
4 .10 120 15
5 .05 120 30

What is the expected profit? (note: profit is defined as the difference between cost and revenue).

Using expected profit as the decision criterion, does it pay for the store to lease a fourth video camera during June at a cost (to them) of $10 a day?

10. A manufacturer of mp3 players has a historical defective rate of 10%. What is the probability that in a batch of 10 players, 3 would be defective?

11. For a binomial distribution, compare P(x = 3) when n = 10 and p = .4 to P(x = 7) when n = 10 and p = .6.

12. Consider a Poisson probability distribution in a process with an average of 3 flaws every 100 feet. Find the probability of no flaws in 100 feet.

13. The weight of a bag of landscape mulch is uniformly distributed over the interval from 38.5 to 41.5 pounds.

What is the probability that a bag will weigh between 39 and 40 pounds?

14. Assume the time it takes to be serviced at a gas station is normally distributed with
 = 3 minutes and  = 0.5 minutes.

What is the probability that service takes between 3.5 and 4.5 minutes?

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