Does the runner-up of four clear the midpoint?

Question

Draw 4 independent numbers uniformly from [0,1] and sort them. What is the probability that the second largest exceeds 0.5?

Accepted Answer

The second largest exceeds 0.5 exactly when at least 2 of the four numbers exceed 0.5 (you need the top two values both above 0.5 for the runner-up to clear it). Each number exceeds 0.5 with probability 0.5, independently, so the count above 0.5 is Binomial(4, 0.5). Take the complement of "fewer than 2": P(	ext{at least } 2) = 1 - \binom{4}{0}\left(	frac12ight)^4 - \binom{4}{1}\left(	frac12ight)^4 = 1 - \frac{1}{16} - \frac{4}{16} = \frac{11}{16} = 0.6875. Takeaway. The j-th largest of a sample exceeds a level t iff at least j values exceed t, turning the question into a binomial count.

Does the runner-up of four clear the midpoint?

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