Solution: Same Data, Either Order

Retrieval check answer. P(neither fails) = 0.97 × 0.97 = 0.9409. P(at least one fails) = 1 − 0.9409 = 0.0591 (about 5.9%) — small individual risks compound faster than intuition suggests once you ask “at least one of several.”

Part 1 — A then B. Beta(2,3) + Batch A (5 succ, 3 fail) → Beta(7, 6). Then Beta(7,6) + Batch B (2 succ, 4 fail) → Beta(9, 10).

Part 2 — B then A. Beta(2,3) + Batch B (2 succ, 4 fail) → Beta(4, 7). Then Beta(4,7) + Batch A (5 succ, 3 fail) → Beta(9, 10). Identical.

Part 3 — Mean: 0.474. 9 / (9+10) = 9/19 ≈ 0.4737. Why order can’t matter: the update rule is nothing but addition — α_new = α_old + successes, β_new = β_old + failures. Whether you add “5 successes, then 2 successes” or “2 successes, then 5 successes” to α, the total is 2 + 5 + 2 = 9 either way — addition doesn’t care what order its terms arrive in. The posterior only ever depends on the total accumulated counts, never the sequence they arrived in. That’s the whole content of “yesterday’s posterior is today’s prior”: every update folds all prior evidence into one distribution, and that distribution is all the next update needs — it carries no memory of how it got there, only how much evidence it represents.

Part 4 — where it breaks. Suppose Batch A was collected from a checkout flow before a pricing change, and Batch B was collected from the same flow after the pricing change shifted the true conversion rate. Combining the counts — Beta(9,10) — silently assumes both batches are evidence about one fixed p, when in fact you have evidence about two different rates that happened to get mixed together. The arithmetic runs identically and produces a posterior that looks just as confident as it would for genuinely stationary data — nothing about Beta(9,10) warns you that the pricing change happened. The fix isn’t “add more data” — more post-change data mixed with pre-change data just compounds the error more confidently. The fix is to recognize the two batches describe different parameters and either model them separately or explicitly track when each observation arrived (a time-varying model), rather than pooling blindly. Order-independence is a genuine, useful property — right up until the thing you’re estimating stops holding still, and then it silently produces a precise-looking wrong answer instead of failing loudly.

The pattern:

Where this goes: next lesson introduces a second conjugate pair — normal-normal — showing that beta-binomial’s “closed-form update by simple addition” isn’t a one-off trick; it’s one instance of a broader pattern called conjugacy, with its own version of “add up the evidence” phrased in terms of precision instead of counts.