• We choose n numbers from a standard normal distribution, and sort them so x

_{1}≤ x

_{2}≤ ⋯ ≤ x

_{n}.

• Then we find the midpoint of each consecutive pair, m

_{i}= (x

_{i}+ x

_{i+1}) / 2.

• These midpoints partition the real line into intervals, one of which, call it J, contains 0.

• If we choose another number from the same normal distribution, it has probability p

_{i}of landing in each interval, which can be expressed with erf().

• Let H be the interval with highest probability (ties are vanishingly rare so we ignore them).

• Let L be the interval with lowest probability.

What are the probabilities that H = J, and that L = J?

In other words, how likely is it that the highest-probability interval contains 0, and how likely is it that the lowest-probability interval contains 0?

I am most interested in the n = 3 case, so I wrote a Monte Carlo program to estimate it, and the results are consistently close to 51.5% for H = J, and 22.2% for L = J. But I’d like to know the exact values, and ideally have an explanation for why.