
Biased coin
One coin is randomly selected from a collection of ten. Nine of these coins are scrupulously fair, but the tenth is biased, landing heads with probability 0.80.
(a) The selected coin is tossed twice, and it lands heads both times. In view of this information, what is the probability that the se lected coin was the biased one?
(b) The selected coin is tossed two more times, again landing heads both times, so we have observed four heads. Now what is the probability that the selected coin was biased?

$\displaystyle \mathcal{H},~\mathcal{B},~ \&~\mathcal{F}$ are the events “two heads, bias coin and fair coin” resp.
$\displaystyle P(\mathcal{H})=P(\mathcal{H}\mathcal{B})+P(\mathca l{H}\mathcal{F})$
$\displaystyle P(\mathcal{B}\mathcal{H})=\frac{ P(\mathcal{H}\mathcal{B})}{ P(\mathcal{H})} $