How the math works

Every pick risks exactly 1 unit at the odds posted when the line went up. A final total above the line wins an Over; below wins an Under; exactly on the line is a push and the unit is refunded.

Win rate ignores pushes: wins ÷ (wins + losses). ROI is units won per unit risked on decided picks.

American odds imply a probability. For negative odds −A it is A ÷ (A + 100); for positive odds +B it is 100 ÷ (B + 100).

That is the win rate needed to break even. Removing the vig from a two-sided market (dividing each side by their sum) gives the book's fair probabilities; at −110/−110 that is 50/50 and the book's hold is 4.55%.

95% range is the Wilson score interval, which stays honest at small sample sizes where the textbook p ± 1.96·SE formula breaks.

Sharp probability is Bayesian: start from a uniform prior on a player's true win rate, update with their wins and losses to get a Beta(W+1, L+1) posterior, and report the probability that it sits above break-even. It's computed exactly with the regularized incomplete beta function.

Majority vote takes whichever side more people picked. Ties sit out. By Condorcet's jury theorem, if voters are independent and each is right more than half the time, the majority gets more accurate as the group grows — but correlated voters (friends who talk it over) add less than they seem to. That's why picks stay hidden until lock.

Skill-weighted uses the optimal rule for independent yes/no voters (Nitzan–Paroush): each vote counts by the log-odds of that person being right.

The 10 phantom wins and losses shrink small samples toward 50%, so a hot 4–0 start barely moves the needle while 60–40 over a hundred picks does. Anyone below 50% gets a negative weight — the model fades them. Weights are rebuilt walk-forward using only results settled before each game started, so the crowd's historical record has no look-ahead.

The fair line is the median of the data rounded to the half point (half the outcomes land on each side). Over probabilities come from two sources: the historical hit rate, and a normal model fitted to the sample mean and standard deviation (with a continuity correction for whole-number totals). Expected value averages the two, and pushes drop out because the stake comes back: