General Probability

Basic Concepts of Probability

Compute the probability that a hand of 13 cards contains

1. the ace and king of at least one suit
2. all 4 of at least 1 of the 13 denominations

A person tried by a 3-judge panel is declared guilty if at least 2 judges cast votes of guilty. Suppose that when the defendant is in fact guilty, each judge will independently vote guilty with probability 0.7, whereas when the defendant is in fact innocent, this probability drops to 0.2. If 70 percent of defendants are guilty, compute the conditional probability that judge number 3 votes guilty given that

1. judges 1 and 2 vote guilty
2. judges 1 and 2 cast 1 guilty and 1 not guilty vote
3. judges 1 and 2 both cast not guilty votes

Let $E_i$, $i = 1, 2, 3$ denote the event that judge $i$ casts a guilty vote. Are these events independent. Are they conditionally independent? Explain.

Numerical Events and Random Variables

Suppose that two defective refrigerators have been included in a shipment of six refrigerators. The buyer begins to test the six refrigerators one at a time. Let the random variable $Y$ represent the number of the test in which the last defective refrigerator is identified. Compute the probabilities for each value of $Y$.

Discrete Mathematics

We have 20 thousand dollars that must be invested among 4 possible opportunities. Each investment must be integral in units of 1 thousand dollars, and there are minimal investments that need to be made if one is to invest in these opportunities. The minimal investments are 2, 2, 3, and 4 thousand dollars. How many different investment strategies are available if

1. an investment must be made in each opportunity?
2. investments must be made in at least 3 of the 4 opportunities?