1989 AHSME Problems/Problem 30
Contents
Problem
Suppose that 7 boys and 13 girls line up in a row. Let be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row we have that . The average value of (if all possible orders of these 20 people are considered) is closest to
Solution 1
We approach this problem using Linearity of Expectation. Consider a pair of two people standing next to each other. Ignoring all other people, the probability that a boy is standing on the left position and a girl is standing on the right position is . Similarly, if a girl is standing on the left position and a boy is standing on the right position the probability is also . Thus, the total probability of the two people being one boy and one girl is .
There are a total of 19 different adjacent pairs, so by Linearity of Expectation, we have that the expected value of is .
Solution 2
Suppose that the class tried every configuration. Boy and girl would stand next to each other in different orders, in different positions, times each. Summing over all gives , so the average value of is .
See also
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