You have a bucket of unfair coins. Calculate the expected value.

Question

I have recently found the following question.

"You have a bucket of unfair coins. Each coin has a probability of getting heads, p, which is uniformly distributed between zero and one. You pick a coin, and flip it 64 times, getting 48 heads. What is the expected value of p for your coin?"

I have no idea how to even approach this question. I tried searching for existing versions of it on StackExchange but could not find anything.

Any help would be much appreciated!

Answer 1

You have $\mathrm p\sim\mathcal U[0,1]$ and $H\mid\mathrm p\sim\mathcal{Bin}(64,\mathrm p)$ and wish to find: $\mathsf E(\mathrm p\mid H{\,=\,}48)$ .

So : -

$$\mathsf E(\mathrm p\mid H{\,=\,}48) = \dfrac{~~\displaystyle\int_0^1 \rho~\mathsf P(H{\,=\,}48\mid\mathrm p{\,=\,}\rho)\,\mathrm d\rho~~}{\displaystyle\int_0^1 \mathsf P(H{\,=\,}48\mid\mathrm p{\,=\,}\rho)\,\mathrm d\rho} $$