2.1.5 problem 7

(a)

$$\begin{array}{cccc}P\left(D\right|H)& =\frac{P\left(D\right)P\left(H\right|D)}{P\left(H\right)}& & \\ & =\frac{P\left(D\right)P\left(H\right|D)}{P\left(D\right)P\left(H\right|D)+P(D^c\left)P\right(H\left|D^c\right)}& & \\ & =\frac{\frac{1}{2}(\frac{1}{100}+\frac{99}{100}{\left(\frac{1}{2}\right)}^7)}{\frac{1}{2}(\frac{1}{100}+\frac{99}{100}{\left(\frac{1}{2}\right)}^7)+\frac{1}{2}{\left(\frac{1}{2}\right)}^7}& & \\ & =0.69& & \end{array}$$

(b)

Let $C$ be the event that the chosen coin is double-headed. $$\begin{array}{cccc}P\left(C\right|H)& =P\left(D\right|H\left)P\right(C|D,H)+P\left(D^c\right|H\left)P\right(C|D^c,H)& & \\ & =0.69\ast 0.56+0& & \\ & =0.39& & \end{array}$$