n men and m women are standing in a line (randomly). Find the expectancy of the number of men that

April Bush

April Bush

Answered question

2022-06-25

n men and m women are standing in a line (randomly).
Find the expectancy of the number of men that stand beside a women (at least one side - left or right)
Harder question: Now solve it, but they stand in a circle and not a line.
I wanted to use an indicator:
X = number of men standing beside at least one women.
X i = 1       if standing besides a women                     0       else
And so: E [ X ] = E [ i = 1 n X i ] = i = 1 n E [ X i ]
the problem is that I am having a time computing what is the probability a random men will stand next to a women (left or right or both)

Answer & Explanation

robegarj

robegarj

Beginner2022-06-26Added 24 answers

E X 1 = P ( X 1 = 1 ) = 1 P ( man 1 has male neighbors ) =
1 ( n 1 2 ) ( m 0 ) ( n + m 1 2 ) = 1 n 1 n + m 1 n 2 n + m 2
Moreover by symmetry we have E X i = E X 1 for every i { 1 , , n } so that:
E X = n E X 1 = n ( 1 n 1 n + m 1 n 2 n + m 2 )
To find E X 1 = P ( X 1 = 1 ) in the first problem is a bit harder.
If E denotes the event then man 1 is at utmost left or utmost right position then we must split up:
We must split up in:
P ( X 1 = 1 ) = P ( X 1 = 1 E ) P ( E ) + P ( X 1 = 1 E ) P ( E )
Here P ( E ) = 2 n + m and P ( X 1 = 1 E ) = m n + m 1
For P ( X 1 = 1 E ) we find what we found for P ( X 1 = 1 ) in the first problem.

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