Probability the vending machine works next time

Question

A vending machine dispensed correctly on 8 of your last 10 purchases. You start from a mild prior $\text{Beta}(2, 2)$ on its true reliability $p$ , centered at one-half, like having already seen 2 good uses and 2 failures.

Using Bayesian updating, what is the probability that your next purchase dispenses correctly?

Accepted Answer

Intuition first. You want the chance the next purchase works, not just the long-run rate. The Bayesian prediction averages the success probability across all plausible values of p, weighted by belief, and that average is the posterior mean. The update rule. Beta is conjugate to works/fails data, so add successes to the first number and failures to the second: 	ext{Beta}(a,b)\ +\ k 	ext{ successes and } (n-k) 	ext{ failures}\ ightarrow\ 	ext{Beta}(a+k,\ b+n-k). Here a=2, b=2, with 8 successes and 2 failures: 	ext{posterior} = 	ext{Beta}(2+8,\ 2+2) = 	ext{Beta}(10, 4). Probability the …

Probability the vending machine works next time

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