3.2.6 problem 22

(a)

Let $C_i$ be the event that $i$-th type of coin is chosen. Let $H_k$ be the event that $k$ out of the $n$ flips land heads.

$$P(X=k)=P\left(C_1\right)P\left(H_k\right|C_1)+P(C_2\left)P\right(H_k\left|C_2\right)=\frac{1}{2}\left(\frac{n}{k}\right)p_1^k{(1-p_1)}^{n-k}+\frac{1}{2}\left(\frac{n}{k}\right)p_2^k{(1-p_2)}^{n-k}$$

(b)

if $p_1=p_2$, then $X$ is Binomial $n$, $k$.

(c)

If $p_1\ne p_2$, then the Bernoulli trials are not independent. If, for instance, $p_1$ is small and $p_2$ is large, and after the first million flips we see two heads, this increases the likelihood that we are using the coin with probability $p_1$ of landing heads, which in turn tells us that subsequent flips are unlikely to be land heads.