(Solution) - Suppose the first trial has probability of success 0 5 and -(2025 Original AI-Free Solution)

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Suppose the first trial has probability of success 0.5, and the second is successful with probability 0.8 if the first is and succeeds with probability 0.2 if the first fails. Show that the second trial has a probability 0.5 of success. Find the probabilities of zero, one, and two successes, and compare with the binomial distribution with n = 2 and p = 0.5. Does the expected number of successes match the binomial distribution? Does the variance?
The binomial distribution depends on two key assumptions: the trials must each have the same probability of success, and the trials must be independent. Show that the distribution of outcomes does not match the binomial distribution in the following cases.