6 men and 4 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 man (so X=1 indicates that a man achieved the highest score on the exam).
a. Find P{X = i}, i=1, 2, 3, ..., 8, 9, 10.

The probability distribution table for X is attached below.

Step-by-step explanation:

Total number of students = 10

All the 10 students gets different ranks from the range 1 to 10.

Number of females = 4

Number of males = 6

The random variable X is defined as the highest ranking achieved by a man.

Total number of ways of ranking 10 different scores is: 10!

Number of ways a male can be ranked 1 is:

= (#ways of choosing any one male out of 6)

                           × (#ways of arranging the rest 9)

= (⁶C₁ × 9!)

Compute the value of P (X = 1) as follows:

Number of ways a male can be ranked 2 is:

= (#ways of choosing any one female out of 4)

                × (#ways of choosing any one male out of 6)

                           × (#ways of arranging the rest 8)

= (⁴C₁ × ⁶C₁ × 8!)

Compute the value of P (X = 2) as follows:

Number of ways a male can be ranked 3 is:

= (#ways of choosing two female out of 4)

                × (#ways of choosing any one male out of 6)

                           × (#ways of arranging the rest 7)

= (⁴C₂ × ⁶C₁ × 7!)

Compute the value of P (X = 3) as follows:

Number of ways a male can be ranked 4 is:

= (#ways of choosing three female out of 4)

                × (#ways of choosing any one male out of 6)

                           × (#ways of arranging the rest 6)

= (⁴C₃ × ⁶C₁ × 6!)

Compute the value of P (X = 4) as follows:

Number of ways a male can be ranked 5 is:

= (#ways of choosing four female out of 4)

                × (#ways of choosing any one male out of 6)

                           × (#ways of arranging the rest 5)

= (⁴C₄ × ⁶C₁ × 5!)

Compute the value of P (X = 5) as follows:

Since there are only 4 female students, the lowest value of X can be 5.

So the probabilities of X greater than 6 are 0.

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