A candidate believes that 2/3 of the registered voters in his district will vote for him in the next election. If two registered voters are independently selected at random, find the probability that neither will vote for him in the next election.
To find the probability that neither of the two selected voters will vote for the candidate, we need to find the probability that each of them individually will not vote for him and then multiply these probabilities together.
Let's break down the problem step by step:
1. The candidate believes that 2/3 of the registered voters in his district will vote for him. This means that the probability a registered voter will vote for him is P(vote for him) = 2/3.
2. The probability that a registered voter will not vote for him is the complement of the probability that they will vote for him. So, P(not vote for him) = 1 - P(vote for him) = 1 - 2/3 = 1/3.
3. Since the two voters are selected independently, the probability that neither of them will vote for him is the product of the individual probabilities. Therefore, P(neither will vote for him) = P(not vote for him) × P(not vote for him).
Now, let's calculate these probabilities:
P(not vote for him) = 1/3
P(neither will vote for him) = (1/3) × (1/3) = 1/9
So, the probability that neither of the two selected voters will vote for the candidate is 1/9.