Probability problems day 8

  1. You’re playing a game of HORSE. You make about 2/3 of your shots. How long could you survive, on average, if your opponent never missed a shot?

  2. Let \(X \sim \text{Poisson}(\lambda)\).

    1. If \(\lambda = 3.5\), what is the probability that \(X = \lambda\)?
    2. What is the probability that \(X = \lambda\) if \(\lambda\) is a positive integer?
    3. If you wanted \(X = \lambda\) to happen, would you want \(\lambda\) to be bigger or smaller? (You should probably use a computer to make a graph.)

  3. Remember the tennis problem? Two people, A and B, are playing a tennis match. With probability \(p\), A wins a point, and with probability \(1 - p\), B wins a point. The match ends the instant someone is up two. Find the expected number of rounds played in the match.

  4. If \(X \sim \text{Poisson}(\lambda)\), is \(X + X\) a Poisson variable?