Standing in line
Let there be people standing in a line, all facing forward and able to see the people in front of them.
How large does have to be for the expected number of people who can correctly assert "I am taller than everyone standing in front of me" to exceed 4?
The answer is 31.
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1 solution
Let us find the general solution for N people.
Let E ( N ) be the expected number of people able to correctly assert the statement. If we remove the last person in line, then we have N − 1 people in line and our expected value is E ( N − 1 ) . Now add back the last person. The chance of them being the tallest person in line is 1 / N , so E ( N ) = E ( N − 1 ) + 1 / N . Clearly E ( 1 ) = 1 , so crank the handle:
E ( N ) = 1 + 1 / 2 + 1 / 3 + . . . + 1 / N .
This is the harmonic series. It first exceeds 4 at N = 3 1