OR 542 Stochastic Models

Problem Set #1 - Probability Review

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.

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.