4.2.3 problem 26

(a)

Let $Z$ represent the number of flips until both Nick and Penny flip Heads. Then is $Z\sim \text{FS}\left(p_1{p}_2\right)$, since Nick’s and Penny’s flips are independent. $\text{E}\left(Z\right)=\frac{1}{p_1{p}_2}$

(b)

The logic is analogous to part $a$, but success probability is $p_1+p_2-p_1{p}_2$.

(c)

$$P(X_1=X_2)=\sum _{k=1}^{\infty }\left({\left({(1-p)}^2\right)}^{k-1}{p}^2\right)=\frac{p}{2-p}$$

(d)

By symmetry, $$P(X_1<X_2)=\frac{1-\left(\frac{p}{2-p}\right)}{2}=\frac{1-p}{2-p}$$