Suppose we have a large pool of biased coins with different Heads-probability; the Heads-probability is a random variable with some complicated distribution (e.g. truncated normal). We want to estimate the probability that, if we draw a coin at random and toss it, it will land on Heads. I consider the following three experiments:
- Draw $n$ coins; toss each coin $m$ times; count the number of Heads and divide by $m n$.
- Draw $m n$ coins; toss each coin one time; count the number of Heads and divide by $m n$.
- Draw $1$ coin; toss it $m n$ times; count the number of Heads and divide by $m n$.
Note: the pool is much larger than $m n$, so the fact that we draw $m n$ coins does not change the distribution (alternatively, we can draw a coin $m n$ times with replacement).
Are these experiments equivalent? If not, which of them would give a better estimate of the actual probability?