Question: A coin-making machine produces pennies. Each penny is manufactured to have a probability $P$ of turning up heads. However, the machine draws $P$ randomly from the uniform distribution on $[0,1]$ so $P$ can differ for each coin produced. A coin pops out of the machine. You flip it once, and it comes up heads. Given this information, what is the (conditional) distribution function $F_{P|H}(p)$ for the probability of a head for that coin (where '$H$' denotes conditioning on the head$)?
I totally have no idea how to start tackling this question.
Any hint is appreciated.
HINT as requested
You need Bayes Theorem for the case of one variable being continuous
Your $P$ is article's $X$ (both continuous), and your $H$ is the article's $Y=y$ (both discrete, or more generally, both are events with non-zero prob).
The article shows how to compute the conditional PDF $f_{P|H}(p)$. If you want the conditional CDF $F_{P|H}(p)$ then simply integrate the PDF.
Hope this helps! Lemme know if you need more details.