Question

4 men and 6 women are ranked according to their scores on an exam. Assume that no two scores are alike, and that all 10! possible rankings are equally likely. Let X denote the highest ranking achieved by a woman (so X = 1 indicates that a woman achieved the highest score on the exam). Find each of the following: P(X = 1) = P(X = 3) = P(X = 4) = P(X = 6) =

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Answer 1

We can approach this problem using combinatorics.

First, we need to find the total number of ways to rank the 4 men and 6 women. This is given by 10! (10 factorial), which is the number of permutations of 10 distinct items.

Next, we need to find the number of ways in which a woman can achieve the highest ranking (x = 1). This can be done by fixing the highest ranking to one of the 6 women, and then permuting the remaining 9 people. This gives us 6*9! ways to arrange the people such that a woman achieves the highest ranking.

Therefore, p(x = 1) = (6*9!)/10! = 6/10 = 0.6

For x = 3, we need to choose 2 women (out of 6) who will get the top 3 rankings, and then permute the remaining 8 people. This gives us (6 choose 2)*8! ways to arrange the people such that 2 women get the top 3 rankings. Therefore, p(x = 3) = [(6 choose 2)*8!]/10! = 15/54 = 0.2778

For x = 4, we need to choose 3 women (out of 6) who will get the top 4 rankings, and then permute the remaining 7 people. This gives us (6 choose 3)*7! ways to arrange the people such that 3 women get the top 4 rankings. Therefore, p(x = 4) = [(6 choose 3)*7!]/10! = 20/54 = 0.3704

For x = 6, all the women will have the lowest rankings. This can be done by permuting the 4 men and then permuting the 6 women. This gives us 4!*6! ways to arrange the people such that all women have the lowest rankings. Therefore, p(x = 6) = (4!*6!)/10! = 0.01296

Note that the sum of probabilities for all possible values of x should be equal to 1.

Explanation: