Answer:
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:
[tex]P(X=1)=\frac{{6\choose 1}\times9!}{10!}=0.60[/tex]
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:
[tex]P(X=2)=\frac{{4\choose 1}\times{6\choose 1}\times8!}{10!}=0.267[/tex]
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:
[tex]P(X=3)=\frac{{4\choose 2}\times{6\choose 1}\times7!}{10!}=0.05[/tex]
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:
[tex]P(X=4)=\frac{{4\choose 3}\times{6\choose 1}\times6!}{10!}=0.005[/tex]
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:
[tex]P(X=5)=\frac{{4\choose 4}\times{6\choose 1}\times5!}{10!}=0.0002[/tex]
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.
