What happens to Kelly when your probability estimate is wrong

If you overestimate your winning probability, Kelly stakes too much, and the damage is lopsided. At even money with a true 55% chance, the Kelly stake is 10%. Believing 60% gives a 20% stake, and at 20% the expected log growth of the bankroll is slightly negative: a real edge turned into a slow loss.

The Kelly criterion calculator shows the stake for any estimate, so you can test how far a wrong number moves it.

The growth rate is a hill with a steep side

The expected log growth per bet at stake f is g(f) = p × ln(1 + b × f) + q × ln(1 − f). It peaks at the Kelly stake, falls slowly on the low side and falls fast on the high side. At the true p = 55%, even money (b = 1):

• Stake 5% (half Kelly): 0.375% growth per bet
• Stake 10% (full Kelly): 0.501% per bet
• Stake 15% (1.5 times Kelly): 0.374% per bet
• Stake 20% (twice Kelly): -0.014% per bet

Half Kelly keeps three quarters of the highest growth. Twice Kelly keeps none of it. The same distance from the peak costs very different amounts depending on which way you miss, and an estimate that is too high misses upward.

The same thing at -110

At -110, b = 0.909 and the break-even rate is 52.38%. Say the true chance is 54%: the Kelly stake is 0.54 − 0.46 ÷ 0.909 = 3.4%, growing the log bankroll by about 0.053% per bet. A bettor who estimates 57% stakes 9.7%, which is 2.85 times too much, and at the true 54% the growth per bet is -0.13%.

Over 1,000 such bets the typical (median) bankroll path is the exponential of the growth sum. The 3.4% stake ends near 1.69 times the starting bankroll. The 9.7% stake ends near 0.27 times it. Both bettors found the same edge, and one went broke slowly because the probability was three points too high.

How wrong is a normal estimate?

If you estimate a win rate from your own record, the standard error is the square root of p × q ÷ n. At a 54% rate it is about 5.0 points after 100 bets and 1.6 points after 1,000. At -110 the Kelly stake moves by (b + 1) ÷ b = 2.1 points of bankroll for each point of probability, so a 1.6-point error moves a 3.4% stake by about 3.3 points, roughly doubling it. Even a thousand bets leave an uncomfortable margin, and a model-based probability has its own error that no sample size removes.

What the arithmetic suggests doing about it

• Stake a fraction of Kelly. The loss from staking below the peak is small, and the fraction acts as a margin for error.
• Round edge estimates down, and treat a zero stake as the answer when the edge is within a point or two of break-even.
• Cap any single stake at a fixed share of bankroll whatever the formula says.
• Write the estimate before the result, so you can compare estimates with outcomes later.

None of these makes the probability right. They limit what a wrong one costs.

Test it. Enter the same price with an honest estimate and an optimistic one in the Kelly criterion calculator and compare the two stakes.