In a group of people, percent have brown eyes. Two people are to be selected at random from the group. What is the probability that neither person selected will have brown eyes?

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The correct answer is A.

One way to determine the probability that neither person selected will have brown eyes is to count both the number of ways to choose two people at random from the people who do not have brown eyes and the number of ways to choose two people at random from all people, and then compute the ratio of those two numbers.

Since percent of the people have brown eyes, there are people with brown eyes, and people who do not have brown eyes. The number of ways of choosing two people, neither of whom has brown eyes, is : there are ways to choose a first person and ways to choose a second person, but there are ways in which that same pair of people could be chosen. Similarly, the number of ways of choosing two people at random from the people is . Therefore, the probability that neither of the two people selected has brown eyes is .

Another way to determine the probability that neither person selected will have brown eyes is to multiply the probability of choosing one of the people who does not have brown eyes at random from the people times the probability of choosing one of the people who does not have brown eyes at random from the remaining people after one of the people who does not have brown eyes has been chosen.

Since percent of the people have brown eyes, the probability of choosing one of the people who does not have brown eyes at random from the people is . If one of the people who does not have brown eyes has been chosen, there remain people who do not have brown eyes out of a total of people; the probability of choosing one of the people who does not have brown eyes at random from the people is . Therefore, if two people are to be selected from the group at random, the probability that neither person selected will have brown eyes is .