Player props,
priced like probabilities.

A batter's per-plate-appearance hit rate is treated as a posterior, not a batting average: a league-and-role prior is updated with the hitter's observed hits and opportunities, so a hot week moves the estimate less than a full season does. That posterior feeds a beta-binomial count distribution over the day's expected plate appearances — which is set only after the lineup slot is known — and each threshold (1+, 2+, 3+ hits) reads its probability straight off that one distribution. Exact-threshold calibration then adjusts each rung of the ladder on its own track record.

Strikeout props are count data with real variance, so the K head blends a pitcher's season rate, recent form, and the opponent's team K rate — each shrunk toward its league prior by sample size — then adjusts for whiff rate, chase rate, expected batters faced, workload, park, and plate umpire. The result is a negative-binomial distribution whose dispersion parameter decides how wide a pitcher's plausible game is. The over/under ladder at 3.5 through 9.5 is priced from that full distribution, which is why two pitchers with the same mean projection can get different prices.

Total bases are a convolution problem: per-plate-appearance rates for outs and walks, singles, doubles, triples, and home runs are combined across the same plate-appearance distribution used by the hits head, producing a probability for every possible bases total. Alternate lines like 2+ total bases are read off the accumulated distribution instead of a normal approximation, so extra-base power shows up as a fat right tail rather than a bumped average.

New model heads do not ship on a hunch. A challenger runs in shadow first — its probabilities are recorded against the same slates without affecting a single published pick — and it can only take over a market when its out-of-sample evidence beats the incumbent's under the monitored promotion gates. Until then the official probability keeps coming from the current head, and the challenger's track record stays diagnostic-only. The family figures above therefore describe heads that survived that process, not whichever idea looked best in a notebook last week.