Sequential Beta updating over three rounds

Question

You estimate a success probability p, starting from a flat 	ext{Beta}(1, 1) prior. Trials arrive in three rounds: - Round 1: 2 successes out of 3 - Round 2: 1 success out of 2 - Round 3: 3 successes out of 4 You update after each round, feeding one round's posterior in as the next round's prior. After all three rounds, what is the probability that the next trial is a success?

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Intuition first. Bayesian updating gives the same belief whether you feed the data in one lump or round by round, since each posterior becomes the next prior. For a Beta with success/failure data, every update adds that round's successes to the first number and failures to the second, so the sequence just accumulates all successes and failures. Update step by step. Beta prior 	ext{Beta}(a,b) plus k successes and m failures gives 	ext{Beta}(a+k, b+m). - Start: 	ext{Beta}(1, 1). - After Round 1 (2 of 3, so 2 successes, 1 failure): 	ext{Beta}(1+2,\ 1+1) = 	ext{Beta}(3, 2). - After Round 2 (…

Sequential Beta updating over three rounds

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