Probstat/homework 1

In these problems, please hand in your paper solutions. Don't just state the answers, but also show your work.

Basic problems

1. Work on problems 1, 3, 5, 7, 9, and 10 in Counting section in week2 practice 1.

2. (From FCP, Ch 2, Problem 13.) We have two bags. Bag A contains 4 red and 3 black balls. Bag B contains 4 red and 6 black balls. If a ball is randomly selected from each urn, what is the probability that the balls will be of the same color?

Theoretical problems

1. For any events A and B, show that if A and B are independent, then A and B^C are also independent.

2. Consider the following experiment: we toss a coin that turns up head with probability p for n times, independently. Prove that the probability that we get exactly k heads is

${n \choose k}p^{k}(1-p)^{n-k}$.

3. We toss a coin that turns up Head with probability p for many rounds independently. What is the probability that you get Head in the k-th round and get Tail in all other rounds 1, 2, ... , k-1?

4. We are tossing N different balls into N bins. Each ball is equally likely to land in any bins. What's the probability that there is no empty bins? (I.e., empty bins are those with no balls landed on.)

Challenging problems

2. A biased coin tuns up head with probability 0.7. The probability that a random coin is biased is 0.1. Let event A_k be the event that you toss the coin k times and the coin turns up all heads. Let event B be the event that the coin is biased. Compute the following conditional probability: P(B), P(B|A₁), P(B|A₂), P(B|A₃), P(B|A₄), and P(B|A₅). Plot the probabilities in a chart where the x-axis shows P(B|A_i), where i ranges from 0 to 5.