Let
denote the event that the two
flips yield the same faces (1 if the same faces occur, 0 if not), so that

For example,

Let
denote the outcome (number of heads) of the next three flips of either
or
. By the law of total probability,


and in particular we have


Then
![P(Y=y)=\begin{cases}\dbinom3y{p_2}^y(1-p_2)^(3-y)(2{p_1}^2-2p_1+1)+\dbinom3y{p_3}^y(1-p_3)^(3-y)(2p_1-2{p_1}^2)&\text{for }y\in\{0,1,2,3\}\\\\0&\text{otherwise}\end{cases}]()
Jack wants to find
for some given
.
a. With
, we have




b. With
, we'd get


