If I get $s$ successes out of $n$ trials in a binomial distribution, what is the probability $p$ of getting a success in each individual trial?

Presumably $p = s/n$, but what if $s = 0$ or $s = n$? Would the probability $p$ really be $0$ or $1$?

That seems far too certain if the sample size $n$ is very small, so is there some correction factor to account for small sample sizes in estimating $p$?

  • $\begingroup$ It depends. What is your loss function, prior distribution, or preferred estimation method? $\endgroup$ – whuber Nov 29 '18 at 0:05
  • $\begingroup$ Uniform prior distribution in $p$ $\endgroup$ – Kelvin Nov 29 '18 at 0:09