Chapter 2.4, Problem 33Q

In an upstate congressional race, the incumbent Republican $\left(R\right)$ is running against a field of three Democrats $\left(D_1,D_2,\text{ and }{D}_3\right)$ seeking the nomination. Political pundits estimate that the probabilities of $D_1$, $D_2$, or $D_3$ winning the primary are 0.35, 0.40, and 0.25, respectively. Further more, results from a variety of polls are suggesting that $R$ would have a 40% chance of defeating $D_1$ in the general election, a 35% chance of defeating $D_2$, and a 60% chance of defeating $D_3$. Assuming all these estimates to be accurate, what are the chances that the Republican will retain his seat?