Discussion
The following is an arrangement of men (M) and women (W), lined up to purchase tickets for a rock concert :
MWMWMMMWMWMMMWWMMMMWW
MWMMMWMMMWWW
What is the expected value of runs for testing the hypothesis that the arrangement is random ?
Answer: C
The expected number of runs is 16.76.
Count the letters in the two printed lines: there are 33 people, \(n_{1} = 20\) men and \(n_{2} = 13\) women. Under randomness the number of runs R has mean
\[ \begin{aligned} E(R) &= \frac{2n_{1}n_{2}}{n_{1} + n_{2}} + 1 \\ &= \frac{2 \times 20 \times 13}{33} + 1 \\ &= 15.76 + 1 = 16.76 \end{aligned} \]
(The observed arrangement has 18 runs, close to this expectation, so the runs test would not reject randomness.)
None of 14.92, 15.88 or 18.20 is the expectation for any split of 33 people into two groups, since each would need \(n_{1}n_{2}\) to be a non-integer. Dropping the +1 gives 15.76, which is not offered either.
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