Chapter 16: Geometric r.v.’s

Scenario: There are repeated independent trials, each resulting in a success or failure, with constant probability of success for each trial. We are counting the number of trials until the first success.

Example 1.   We throw darts at a dartboard until we hit the bullseye. Assume throws are independent and the probability of hitting the bullseye is 0.01 for each throw.

  1. What is the pmf for the number of throws until we hit the bullseye?

    Solution:

  2. What are the mean and variance for the number of throws until we hit the bullseye?

    Solution:

  3. Find the probability that our first bullseye:

    1. is on the fourth try

      Solution:

    2. is on one of the first four tries

      Solution:

    3. is after the fifth try

      Solution:

    4. is on one of the first fifty tries

      Solution:

    5. is after the \(50^{th}\) try, given that it did not happen on the first 20 tries.

      Solution:

  4. Find the expected number of misses until we hit the bullseye.

    Solution: