We live in a simulation.

For each day, the probability a machine fails is $p$. Given a binary string of $d$ days of data, where $1$ is a failure on that day, estimate the probability $p'$ of $5$ of these machines failing on the same day.

Define a program that takes in an string representing d days of data from one machine, and outputs the optimal unbiased estimation for p', where "optimal" means the estimator has the minimum variance among all valid unbiased estimators.

Answers within $1e{-5}$ of the expected answer will be accepted. In other words, do not worry about floating point truncations or rounding offsets.

Notes

• Assume machine failures are independent across machines
• The estimator should be a function of the observed data only (no simulation/randomization)