Expected Value and Kelly Bet Sizing

Expected value asks whether a price is favorable given your probability estimate. The Kelly Criterion asks how much of a bankroll would maximize long-run logarithmic growth if that estimate were correct.

Both calculations depend on the probability input. Precise arithmetic cannot rescue an inaccurate model.

Expected value for a standard stake

For a $100 stake:

$EV = P(win) \times profit - P(loss) \times 100$

Suppose an underdog is +150 and your documented win probability is 45%.

$EV = 0.45 \times 150 - 0.55 \times 100 = 12.50$

The estimated expected value is +$12.50 per $100 staked, or +12.5%. This is an average across repeated comparable bets, not a prediction that this one bet will earn $12.50.

At +150, the break-even probability is 40%. If your estimate falls to 40%, expected value is zero before other costs. Below 40%, it is negative.

Convert American odds to net decimal odds

Kelly uses net decimal odds, often written as $b$.

For positive American odds:

$b = \frac{odds}{100}$

For negative American odds:

$b = \frac{100}{|odds|}$

At +150, $b = 1.5$. At -150, $b = 0.6667$.

Full Kelly

The two-outcome Kelly fraction is:

$f^* = \frac{bp - q}{b}$

where $p$ is your win probability, $q = 1-p$, and $b$ is the net decimal profit per unit staked.

For +150 with a 45% win estimate:

$f^* = \frac{1.5(0.45) - 0.55}{1.5} = 0.0833$

Full Kelly would allocate 8.33% of bankroll. Quarter Kelly would allocate about 2.08%.

Why fractional Kelly is usually more defensible

Kelly assumes the input probability is correct and the opportunity can be repeated under the same conditions. Real estimates have error. Fractional Kelly reduces the damage when confidence is overstated.

The sensitivity is easy to see. Keep the price at +150 but lower the win estimate from 45% to 42%:

$f^* = \frac{1.5(0.42) - 0.58}{1.5} = 0.0333$

Full Kelly falls from 8.33% to 3.33%. Quarter Kelly falls from 2.08% to 0.83%. A three-point probability revision changes the suggested stake sharply.

When the answer is zero

If $bp-q$ is zero or negative, there is no positive Kelly allocation. Treat the result as no bet. Do not convert a negative value into a small positive stake because you want action on the game.

Multiple bets on one slate

The simple formula treats each wager in isolation. Bets can share risk through the same team, player status, or game environment. Adding several individually calculated Kelly stakes can create more exposure than the bankroll plan intended.

A practical process is to:

1. calculate each bet from the price actually available;
2. use a fixed Kelly fraction consistently;
3. cap total slate exposure before seeing the day's recommendations;
4. record correlated positions together;
5. judge calibration and net returns on future bets, not selected winners.

Connect the calculation to the daily board

Use the odds calculator to convert prices and remove vig. Then open today's NBA board for the current slate and matchup context.

An edge badge is a model estimate relative to a price. It is not a guarantee, and it is not a substitute for checking that the displayed price is still available. Only risk money you can afford to lose.