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Module 3 · Joint distribution of (winner, length)

Eight cells of closed-form probability

For every (winner, series length) pair the posterior predictive probability is a ratio of Beta functions, and the eight cells together sum to one. Marginalise across rows for the winner distribution; across columns for the length distribution. The condition-on-length toggle slices the joint along a single column to show how the winner probability shifts once N is fixed.

the closed-form jointwhy N=7 pulls toward 1/2