STAT/MATH 394B

Instructions:
Do as many problems as you have time for. Give full explanations of your solution. Just an answer is not acceptable.

1. (a) How many integers between 100 and 999 have distinct digits?

(b) How many of the integers with distinct digits in (a) are odd?

2. Suppose that each of ten sticks is broken into a long part and a short part. The 20 parts are arranged into 10 pairs and glued back together, so that again there are ten sticks. What is the probability that each long part will be paired with a short part?

Note: This problem is a model for the effect of radiation on a living cell. Each chromosome subjected to radiation breaks into two parts, one containing the centromere. The cell will die unless the part containing the centromere recombines with one not containing a centromere.

3. Let S={e₁, e₂, ... } be a countable sample space, and introduce the events A_k={e_j: j≥k}. Let P be a probability on S such that kP(A_k) does not depend on k. Show that P is uniquely determined by this condition.

For extra credit:

Let B={e _2j+1 : j=0,1,2, ... }. Find P(B).

4. Someone rolls a die. If he gets a 6, he rolls again, otherwise he rolls twice more. Find the probability that he (in the two or three rolls) gets only even results.

5. Roughly speaking, summertime weather is of one of the following three types:

A₁: high pressure A₂: unstable A₃: low pressure

The different types occur 20, 50 and 30% of the time, respectively. The probability of rain in A₁ weather is 0.05, in A₂ weather 0.4 and in A₃ weather 0.9. Suppose it rains. What is the conditional probability that the weather is of type A_k, k=1,2,3?