Calculating the Probability of Changeovers in Coin Flipping Experiments

Consider n independent flips of a coin having probability p of landing heads. Say a changeover occurs whenever an outcome differs from the one preceding it. For instance, if the results of the flips are HHTHTHHT, then there are a total of five changeovers. If p = 1/2, what is the probability there are k changeovers? 

Let H be the event that a coin flip results in heads, and let T be the event that a coin flip results in tails. Suppose we flip a coin n times and let S be the sequence of outcomes. A changeover occurs whenever the outcome of a flip differs from the one preceding it. If we denote the number of changeovers in S by C, then we can see that:

C = 0 if S consists of all heads or all tails

C = 1 if S consists of alternating heads and tails

C = x if S consists of x + 1 runs of consecutive heads or tails

Since the probability of getting a head in any one flip is p, the probability of getting a tail is 1 - p. Suppose we are given that p = 1/2. Then the probability of getting a head and a tail in any order is 1/2 * 1/2 = 1/4. This means that the probability of getting a run of consecutive heads or tails of length k is (1/2)^(k+1).

We can now calculate the probability that there are k changeovers in n flips. Let C denote the event that there are k changeovers in n flips. Then, we can express this probability as follows:

P(C) = P(C | S starts with H)P(S starts with H) + P(C | S starts with T)P(S starts with T)

where P(S starts with H) = p and P(S starts with T) = 1-p.

We can then calculate P(C | S starts with H) and P(C | S starts with T) separately. Let R denote the number of runs of consecutive heads or tails in S. Then:

P(C | S starts with H) = P(R = k-1) = (n-1 choose k-1) * (1/2)^(n-k)

P(C | S starts with T) = P(R = k-1) = (n-1 choose k-1) * (1/2)^(n-k)

Therefore, we have:

P(C) = (n-1 choose k-1) * (1/2)^(n-1)

This formula shows that the probability of k changeovers in n flips of a fair coin is given by the binomial distribution with parameters n-1 and 1/2.

We have shown that the probability of changeovers occurring in coin flipping experiments can be calculated using the binomial distribution. Specifically, if we flip a coin n times and let S be the sequence of outcomes, we can calculate the probability of k changeovers in S using the formula (n-1 choose k-1) * (1/2)^(n-1). This problem involves understanding the binomial distribution, combinatorics, and the theory of independent events. By following the steps outlined above, we can solve similar probability problems involving changeovers in coin flipping experiments.