Chapter 8.6, Problem 53E

To determine

To find:the probability that the incumbent wins the election.

Answer to Problem 53E

The probability of winning one incumbent politician= $\frac{6}{8}$.

Explanation of Solution

Given information:The incumbent and two first-time politicians are the only candidates

for a public office.The first-time politicians have the same probability of winning. The incumbent has six times the probability of winning as either of the other candidates

Formula Used:

If an event has n(E) equally likely outcomes and its sample space S has n(S) equally likely outcomes, then the probability of event is given by,

  $P(E)=\frac{n(E)}{n(S)}.$

Calculation:

Total number of candidate =one incumbent politician + two first-time politician=1+2=3.

Let chance of winning one first-time politician is = a.

Hence the probability of winning first first-time politician= $\frac{a}{3}$ .

The probability of winning second first-time politician= $\frac{a}{3}$ .

The probability of winning one incumbentpolitician=6 times $\frac{a}{3}$ = $\frac{6a}{3}$ .

Hence, $\frac{a}{3}+\frac{a}{3}+\frac{6a}{3}=1\Rightarrow 8a=3\Rightarrow a=\frac{3}{8}.$

Hence the probability of winning first first-time politician= $\frac{1}{8}$.

The probability of winning second first-time politician= $\frac{1}{8}$.

The probability of winning one incumbent politician= $\frac{6}{8}$.