Five goals, all in the first half?

Question

Goals in a 90-minute soccer match arrive as a Poisson process. The match ends with exactly 5 goals. Given this, what is the probability that all 5 goals were scored in the first half (the first 45 minutes)?

Accepted Answer

Conditioned on the total number of goals, each goal's time is an independent uniform point over the 90-minute match; the rate no longer matters. So each of the 5 goals independently fell in the first half with probability \frac{45}{90} = \frac12. For all 5 to land in the first half, we need this to happen five times independently: \left(\frac12\right)^5 = \frac{1}{32} \approx 0.03125. So only about a 3\% chance. The Poisson rate never entered; conditioning on "5 goals" reduced the problem to five uniform coin-flips over halves. Takeaway. Given the count, each arrival independently lands in a …

Five goals, all in the first half?

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