# comparison of two biased coins, flipped by biased observers

We have 2 coins that we don't know whether they are biased or not. M observers, each flip each coin N times and get fraction of heads to total flips of $$p_m^1$$ and $$p_m^2$$. Therefore, for each coin i we have:

$$\begin{eqnarray*} P_i = \frac{\sum_{m = 1}^{M}p_m^i}{M} \end{eqnarray*}$$

However, each observer has biased fingers (in other words, even if the coins are fair they would get head more than tails, for example).

How can we test whether $$P_1$$ and $$P_2$$ are statistically different from each other? And what is the SEM for each coin?