Question

3 men and 7 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). Find each of the following:
P(X=2)=
P(X=6)=
P(X=7)=

Answer 1

3 men and 7 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).
total number of members (men and women) =10
probability of chance (men and women) is 1/2 =0.5

BIONOMIAL DISTRIBUTION
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is executed
p = success probability
mean = 10 * 0.5
= 5
II.
variance = npq
where
n = total number of repetitions experiment is executed
p = success probability
q = failure probability
variance = 10 * 0.5 * 0.5
= 2.5
III.
standard deviation = sqrt( variance ) = sqrt(2.5)
=1.5811

a.
P( X = 2 ) = ( 10 2 ) * ( 0.5^2) * ( 1 - 0.5 )^8
= 0.0439

b.
P( X = 6 ) = ( 10 6 ) * ( 0.5^6) * ( 1 - 0.5 )^4
= 0.2051

c.
P( X = 7 ) = ( 10 7 ) * ( 0.5^7) * ( 1 - 0.5 )^3
= 0.1172