Bet Size Optimizer

The Bet Size Optimizer sizes a stake from the Kelly criterion, then multiplies by a fraction you choose. Four inputs: Bankroll (default $1000), Win Probability (default 55%), Decimal Odds (default 1.91), Kelly Fraction (default 50%). Full Kelly f* = (b p − q) / b with b = odds − 1, p = win probability, q = 1 − p. Applied fraction = f* × (Kelly Fraction / 100). Optimal Stake = bankroll × applied. On the defaults, b = 0.91, p = 0.55, q = 0.45, b p − q = 0.5005 − 0.45 = 0.0505, f* = 0.0505/0.91 ≈ 0.0554945 → Full Kelly 5.55%. Half Kelly: applied = 2.7747%, stake = $27.75.

The $27.75 is 2.77% of a $1000 roll at a 55% shot on 1.91. Full Kelly would stake $55.49. The default 50% fraction is the common “half Kelly” compromise: less variance, less growth, fewer ruin paths. Set Kelly Fraction to 100 to see full Kelly; set it to 25 for quarter Kelly ($13.87). Set Win Probability to 52% at the same 1.91: f* = (0.91×0.52 − 0.48)/0.91 = (0.4732 − 0.48)/0.91 < 0 → stake $0.00. The edge vanished; Kelly is zero. That is the optimizer doing its job.

Odds 1.91 is the classic ~1/1.91 ≈ 52.36% implied probability; your 55% is a 2.64 percentage-point edge. Raise odds to 2.00 at 55%: b = 1, f* = 0.55×1 − 0.45 = 0.10, 10% full Kelly, $50 at half Kelly. Lower odds to 1.80 at 55%: b = 0.80, b p − q = 0.44 − 0.45 < 0, again zero. Price and probability have to fit.

The page does not know your real p. A 55% that is actually 51% turns a “half Kelly $27.75” into an overbet. Fractional Kelly is a hedge against that estimation error, which is why Thorp and later sports syndicates default to a fraction. Read Full Kelly % and Bankroll % together: 5.55% and 2.77% on the defaults. If $27.75 exceeds a book limit, the math is unchanged and the constraint is elsewhere.

John L. Kelly Jr.’s 1956 Bell Labs paper, A New Interpretation of Information Rate, derived a stake that maximizes the expected logarithmic growth of wealth in repeated bets. Edward O. Thorp carried Kelly into blackjack and the stock market; sports bettors adopted it because decimal odds make b = decimal − 1 obvious. Full Kelly is aggressive: it maximizes E[log W] and produces a volatile wealth path. Half Kelly, quarter Kelly, and other fractions are engineering responses to uncertainty in p, to non-Kelly utility, and to the fact that the discrete bet is not the continuous-time Brownian model in which Kelly is uniquely optimal.

SorteCalc applies the standard binary-bet formula f* = (b p − q)/b, floors it at 0 (no negative bets), then scales by the fraction percent. It is not a horse-racing simultaneous Kelly (multiple correlated outcomes), not a proportional Kelly across a coupon of parlays, not a market-making inventory model. One bet, two outcomes, known odds, estimated p.

The 1.91 / 55% / $1000 / 50% default is the sports-book classroom: a small edge on a standard decimal price, half Kelly. Casino games with negative edge yield f* = 0; the optimizer will print $0.00 if you type 48% on 2.00. That is a feature. Kelly does not “find a stake” for a bad bet; it finds zero.

Limits: p known, odds fixed, even-money-or-better binary payoff, no simultaneous bets, no drawdown constraint, no tax, no commission (Betfair 2–5% would need a net-odds adjustment). Overestimating p is the main way Kelly blows up; the log-optimal stake on a wrong p is not log-optimal on the true p. Literature (MacLean, Thorp, Ziemba) spends chapters on fractional Kelly for that reason. A 50% fraction is not magic; it is a widely used knob.

Use the tool to convert an edge into a percent of bankroll, to see the cliff where f* hits 0, and to compare full versus half. Do not treat $27.75 as a bookmaker’s invitation. Kelly assumes you can lose the stake and continue with 97.23% of the roll. If that is false, your utility is not log.

A lecture on how to size a stake from an estimated edge begins with Kelly’s growth fraction, then scales it by a confidence you choose. John L. Kelly Jr.’s 1956 Bell Labs paper derived the fraction of wealth that maximises expected logarithmic growth on a repeated binary bet. For decimal odds o the net odds are b = o − 1. With win chance p and q = 1 − p, the growth-optimal fraction is the maximum of zero and (b p − q)/b. Defaults: bankroll 1000, p = 55 percent, o = 1.91, confidence — the applied fraction of Kelly — 50 percent. Then b = 0.91, b p − q = 0.91 × 0.55 − 0.45 = 0.0505, full fraction = 0.0505/0.91 ≈ 0.0554945, or 5.55 percent of bankroll.

Stake is bankroll times that edge-like fraction times confidence over 100: 1000 × (5.55/100) × (50/100) ≈ 27.75. Equivalently, half of the 5.55 percent is 2.77 percent of 1000, which is 27.75. Full Kelly would stake about 55.49. Quarter Kelly at 25 percent confidence would stake about 13.87. If p = 52 percent at the same 1.91, b p − q = 0.91 × 0.52 − 0.48 is negative, and the stake is zero: there is no edge, so the optimiser refuses to invent one. Raise odds to 2.00 at 55 percent: b = 1, full fraction = 0.10, half Kelly = 50. Lower odds to 1.80 at 55 percent and the edge vanishes again. Price and probability have to fit.

Derivation: maximise p log(1 + f b) + q log(1 − f). Set the derivative to zero and rearrange to f = (b p − q)/b. If the numerator is negative the maximum on the unit interval is f = 0. Implied probability of 1.91 is 1/1.91 ≈ 52.356 percent; your 55 percent is 2.64 points of probability edge. Expected value per unit staked is 0.0505, and 0.0554945 × 0.91 = 0.0505 is the check. Edward O. Thorp carried Kelly into blackjack and markets; MacLean, Thorp, and Ziemba wrote at length on fractional Kelly as a hedge against a misestimated p. A 55 percent that is actually 51 percent turns 27.75 into an overbet; the confidence knob is a poor man’s shrink.

Assumptions: known constant p, known b, log utility, infinitely divisible stakes, independent repeated bets, no limit, no commission. Simultaneous correlated bets need a different vector formula. Casino games with negative edge yield zero, which is a feature. Half Kelly is not the maximiser of a simple closed-form growth rate except “half the log-optimal f”; it is engineering. Kelly assumes you can lose the 27.75 and continue with 97.23 percent of the roll. If that is false, your utility is not log. The 27.75 is 1000 × 0.0555 × 0.50, the product the page is built to teach. A 1.91 book limit that caps you below 27.75 is a constraint elsewhere; the identity does not change.