Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are ali

Five men and 5 women are ranked according to their scores on an examination. Assume that
no two scores are alike and all 10! possible rankings are equally likely. Let X denote the
highest ranking achieved by a woman (for instance, X=1 if the top-ranked person is female).
Compute the expected value of the random variable

Miguel -

Expected Value = 11/6

As you correctly noted, there are 10! rankings in the sample space. So, if done correctly, the sum of the possibilities for each ranking should add up to 10!

Rank 1: 5 x 9!
Rank 2: 5 x 5 x 8!
Rank 3: 5 x 4 x 5 x 7!
Rank 4: 5 x 4 x 3 x 5 x 6!
Rank 5: 5 x 4 x 3 x 2 x 5 x 5!
Rank 6: 5 x 4 x 3 x 2 x 1 x 5 x 4!

If you add these up it will equal 10!

So, the probability for each rank is simply the value above divided by 10!

Expected value is simply the rank number above times the respective probabilities.

Answer is 11/6

Hope that helps

Probability position 1 is a woman: 5/10 or 1/2.

Probability position 1 is a man and position 2 is a woman: 5/10 * 5/9 = 5/18

Probability positions 1 and 2 are men and position 3 is a woman: 5/10 * 4/9 * 5/8 = 5/36

Probability positions 1 through 3 are men and position 4 is a woman: 5/10 * 4/9 * 3/8 * 5/7 = 5/84

Probability positions 1 through 4 are men and position 5 is a woman: 5/10 * 4/9 * 3/8 * 2/7 * 5/6 = 5/252

Probability positions 1 through 5 are men and position 6 is a woman: 5/10 * 4/9 * 3/8 * 2/7 * 1/6 * 5/5 = 1/252

The expected value is 1 * 1/2 + 2 * 5/18 + 3 * 5/36 + 4 * 5/84 + 5 * 5/252 + 6 * 1/252 = 11/6