Kelly Criterion for Sports Betting: Bet Sizing Math

The formula

f* = (bp − q) / b

• f* is the share of your bankroll to stake.
• b is the net decimal odds: decimal odds minus 1, or the profit on a $1 stake. For a negative American price A (drop the minus), b = 100 / A. For a positive price A, b = A / 100.
• p is the chance you believe the bet wins.
• q is the chance it loses, 1 − p.

The top of the fraction, bp − q, is the expected value of a $1 stake. So Kelly is simply your edge divided by the net odds. A bigger edge means a bigger stake; a longer price means a smaller share for the same edge.

Worked example: -110 at a 55% chance

You believe a side priced at -110 wins 55% of the time. The bankroll is $1,000.

• b = 100 / 110 = 10 / 11 (decimal odds 21 / 11, about 1.91)
• bp − q = (10 / 11 × 0.55) − 0.45 = 0.50 − 0.45 = 0.05
• f* = 0.05 / (10 / 11) = 0.055, which is 5.50% of the bankroll

Full Kelly stakes $55.00. The 0.05 edge is the same +$5.00 per $100 found in the expected value article for this exact bet; Kelly turns it into a stake size. If the bet wins, the bankroll goes to $1,050.00; if it loses, to $945.00. Either way, the next stake is 5.50% of the new total.

Worked example: a plus price

A side is posted at +150 and you believe it wins 42% of the time. Break-even at +150 is 40.00%.

• b = 150 / 100 = 1.5
• bp − q = (1.5 × 0.42) − 0.58 = 0.63 − 0.58 = 0.05
• f* = 0.05 / 1.5 = 1 / 30, about 3.33%

On $1,000 that is $33.33. The edge per $1 is the same 0.05 as the first example, but the stake is smaller, because a longer price loses more often and swings the bankroll harder.

Worked example: using a fair price as p

The hardest input is p. One grounded source is the market itself. In no-vig fair odds, a game posted Team A -140 / Team B +120 gives Team B a no-vig chance of 60 / 137, about 43.80%. Suppose another book has Team B at +135.

• b = 1.35
• bp − q = (1.35 × 60 / 137) − 77 / 137 = (81 − 77) / 137 = 4 / 137
• f* = (4 / 137) / 1.35 = 80 / 3,699, about 2.16%

That is $21.63 on $1,000 at full Kelly, $10.81 at half, $5.41 at quarter. A small price edge earns a small stake, which is the point.

No edge means no bet

At -110 the break-even chance is 11 / 21, about 52.38%. Plug that in as p and bp − q is exactly 0, so f* = 0. At a 50% chance the formula returns −5.00%. A negative answer means the price is minus-EV at your estimate; the stake is zero.

That makes Kelly a useful filter even if you never use it to size: if you cannot show bp − q above zero with honest inputs, the bet does not belong on the card. For the break-even rates behind this, see implied probability and the vig.

Half Kelly and quarter Kelly

Fractional Kelly stakes a fixed share of the full Kelly amount. Half Kelly on the -110 example is 2.75% ($27.50); quarter Kelly is 1.375% ($13.75).

Full Kelly is the stake that gives the fastest long-run growth rate when p is exactly right. The growth rate per bet is p × ln(1 + bf) + q × ln(1 − f), the average log change in the bankroll. Here is how it moves with stake size, assuming the 55% chance is true:

Half Kelly keeps about three quarters of the growth rate for half the stake. Double Kelly, on a bet that is truly plus-EV, has a growth rate just below zero, and triple Kelly shrinks the bankroll steadily. Swings drop too: five straight losses take $1,000 to $753.63 at full Kelly, $869.86 at half, and $933.11 at quarter.

Quarter Kelly here lands at 1.375%, inside the 1–2% unit range covered in bankroll management and unit sizing. That overlap is a good sanity check: if a Kelly fraction asks for far more than your unit rule, look hard at the inputs.

Why overestimating your edge is costly

Kelly is only as good as p. Say you believe the -110 side wins 55%, but the true chance is 53%. The bet is still plus-EV, since 53% is above the 52.38% break-even. Look at what each stake does at the true 53%:

A two-point overestimate makes the full Kelly stake about 4.2 times the right one, and the growth rate turns negative: the bankroll tends to shrink even though every bet is plus-EV. Quarter Kelly on the same mistaken belief lands almost exactly on the right stake.

That is the real case for fractional Kelly. Estimates are noisy, and the cost of betting too much is much steeper than the cost of betting too little. Two more habits help: when several bets are open at once, scale each stake down so the total stays sensible, and grade your estimates over time with closing line value and a bet log.

Put Kelly to work

Start from the price: the American odds implied probability table gives the break-even rate for each posted number, and line shopping raises b for the same bet. Then cap the stake with a fraction of Kelly and your unit rule, and record it. The free bankroll and line-shopping worksheet has room for the price, your stake and the bankroll after each result.

For how bet sizing, bankroll and record keeping fit together into one system, see the book.

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