**OR 542 Stochastic Models**

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**Problem Set #1 - Probability Review**

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1. If we draw one card from a 52 card deck, what is the probability that it is red? That it is a diamond? That it is an ace? That it is the ace of diamonds?

2. A pair of dice is rolled once. Compute the probability that the sum is equal to each of the integers 2 through 12.

3. A fair coin is flipped 4 times. What is the probability that the fourth flip is heads given that each of the first 3 flips resulted in heads?

4. For a particular gambling game in which you bet $1, the following are the payoffs you could get, together with their associated probabilities:

__Payoff__ __Probabilities__

- 1 125/216,

1 75/216,

2 15/216,

3 1/216.

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Compute the expected payoff, and the variance of the payoff.

5. Using the results of problem 2, compute the expected sum of a single roll of 2 fair dice.

6. Box 1 contains 4 defective and 16 nondefective light bulbs. Box 2 contains 1 defective and 1 nondefective light bulb. We roll a fair die 1 time. If we get a 1 or a 2, then we select a bulb at random from box 1. Otherwise we select a bulb from box 2. What is the probability that the selected bulb will be defective?

7. Suppose there is a medical test that diagnoses a particular disease with 95% accuracy on both those that do have the disease and on those that do not have it. If 0.5% of the population actually has the disease, compute the probability that a particular individual has the disease, given that the test says he does.