A single trial has probability p of success and (1-p) of failure. For simplicity, we assume p = 1/2 = (1-p). An experiment is defined as a sequences of trials until 4 consecutive success or failures.
Suppose two such experiments were conducted. What is the probability that they will be identical?
Attempt
Given an outcome of experiment 1 that is made up of k trials. Then the probability of experiment 2 being exactly the same is, $ P(k)(1/2^k) $
where P(k) is the probability of an experiment being k trials in length. Since experiment 1 can be of length 4 to infinity, the final probability is, $\sum_{k=4}^{\infty} P(k)^2(1/2^k)$
Question
- How do I find P(k)?
- Is this the right approach? Or is there a simpler way?