Kelly and expected value: what a probability assumption changes

Why this matters

Kelly arithmetic starts with an assumed probability. The calculator does not supply or validate that probability. Its modelled fraction can change sharply when the input moves by a few percentage points, even though the decimal price stays unchanged.

Before entering the numbers

Kelly arithmetic starts with an assumed probability. The calculator does not supply or validate that probability. Its modelled fraction can change sharply when the input moves by a few percentage points, even though the decimal price stays unchanged.

Open the calculator: Kelly Criterion / Staking Calculator. All figures below are hypothetical. CU denotes a consistent currency unit, not a named currency. Field and result names match the English calculator interface.

Example 1: the starting assumptions

At odds of 2.20 and an assumed probability of 50%, compare the raw fraction with the half-Kelly setting. The 1,000 CU bank and 10 CU entered stake serve different calculations: the modelled amount uses the bank; expected value uses the stake.

Entered fields: Decimal odds: 2.2 · Your estimated win probability: 50 (%) · Illustrative bankroll: 1000 (CU) · Stake for EV calculation: 10 (CU) · Kelly fraction: Half

Calculated results: Raw Kelly fraction: 8.33 % · Modelled fraction: 4.17 % · Modelled amount (not advice): 41.67 CU · Expected net value of entered stake: 1.00 CU · Modelled expected return on stake: 10.00 %

• f = (0.50 × 2.20 − 1) / (2.20 − 1) = 0.083333
• half fraction = 0.083333 × 0.5; amount = 1,000 × 0.041667
• EV = 10 × (0.50 × 2.20 − 1) = 1 CU

Example 2: change the inputs

Lower only the assumed probability to 45%. The raw fraction becomes negative. The interface floors the modelled fraction at zero, while retaining the negative raw value and expected value so the input sensitivity remains visible.

Entered fields: Decimal odds: 2.2 · Your estimated win probability: 45 (%) · Illustrative bankroll: 1000 (CU) · Stake for EV calculation: 10 (CU) · Kelly fraction: Half

Calculated results: Raw Kelly fraction: -0.83 % · Modelled fraction: 0.00 % · Modelled amount (not advice): 0.00 CU · Expected net value of entered stake: -0.10 CU · Modelled expected return on stake: -1.00 %

• f = (0.45 × 2.20 − 1) / 1.20 = −0.008333
• modelled fraction = max(0, f) × 0.5 = 0
• EV = 10 × (0.45 × 2.20 − 1) = −0.10 CU

What the result leaves out

These formulas assume a probability that is known well enough to use in the model. They do not measure estimation error, correlated exposure, drawdown, affordability or an appropriate personal stake.

Arithmetic uses unrounded intermediate values; displayed numbers are rounded. These examples document the model, not a provider quote or an observed transaction.

18+. Educational information only. This is not financial, investment, legal or tax advice, and it is not a guarantee of any result. Betting, gambling and crypto involve a risk of losing money. Estimates depend on your inputs; verify current provider terms and local rules.

Practical checklist

• Record the source, units and time of each input.
• Keep fees and principal in their stated units.
• Reproduce both examples before comparing other inputs.
• Read the stated limitations and current provider terms.