Probability the sensor fires correctly again

Question

A safety sensor has fired correctly on all 5 of its 5 tests so far. You start from a flat prior $\text{Beta}(1, 1)$ on its true reliability $p$ , treating every value between 0 and 1 as equally plausible beforehand.

Using Bayesian updating, what is the probability that the sensor fires correctly on the next test?

Accepted Answer

Intuition first. A flawless run is encouraging, but 5 for 5 does not mean the sensor is literally perfect. The Bayesian prediction averages the success probability across all still-plausible values of p, weighted by belief. That average is the posterior mean, so it stays a shade below 1 to respect the small sample. The update rule. Beta is conjugate to pass/fail data, so add passes to the first number and fails to the second: 	ext{Beta}(a,b)\ +\ k 	ext{ passes and } (n-k) 	ext{ fails}\ ightarrow\ 	ext{Beta}(a+k,\ b+n-k). Here a=1, b=1, with 5 passes and 0 fails: 	ext{posterior} = 	ext…

Probability the sensor fires correctly again

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