What price do you need if you win 55% of the time?

The price that breaks even at a given win chance is 1 ÷ the chance in decimal odds. A 55% chance is 1.818, which is -122 in American odds. Any price longer than -122 has a positive expected value if the 55% is true, and any shorter price has a negative one. The figure applies to a chance you have estimated, not one you know.

Enter a win rate or a price in the break-even calculator to convert in either direction.

The formulas

Decimal break-even price = 1 ÷ p. American, for p over 50%: -100 × p ÷ (1 − p). For p under 50%: +100 × (1 − p) ÷ p. At 55% the first gives -100 × 0.55 ÷ 0.45 = -122.2. At 40% the second gives +100 × 0.60 ÷ 0.40 = +150.

A table of fair prices

• 33.3%: decimal 3.00, +200
• 40%: 2.50, +150
• 45%: 2.22, +122
• 50%: 2.00, +100
• 52.38%: 1.909, -110
• 55%: 1.818, -122
• 60%: 1.667, -150
• 65%: 1.538, -186

The prices on this list are the break-even points. A price offered above one of them for the same chance has a positive expected value, and a price below has a negative one. The list also shows why -110 needs 52.38% rather than 50%: a coin-flip chance is fairly priced at +100, and -110 is a worse price.

The idea behind it

A price is a statement about the chance at which a bet is fair. To compare it with a belief, convert one into the other's unit. Either turn the price into a probability, as in converting a negative price and converting a positive one, or turn the chance into a price as above. The result is the same comparison. Doing it in price terms is useful when you are shopping lines, since the question becomes "is the price on offer longer than my fair price?"

The margin sits between the two prices

The two sides of a market at -110 each imply 52.38% apiece, which add to 104.76%. Your fair price for one side is the price with the margin removed and your own chance in its place. The distance between your fair price and the one on offer is where the expected value lives. If you believe 52% on a -110 side, the fair price is -108 and the offered -110 is slightly short of it.

Why the belief is the weak point

A fair price built on a chance you have invented is only as sound as the chance. Win rates taken from your own record have wide error, as shown in the Kelly estimate article: about 5 points of standard error after 100 bets. A price edge of 2 or 3 points is small against that. The arithmetic here sets the target. Whether your 55% is real is a separate question the calculator cannot answer.

Convert yours. The break-even calculator returns the fair price for any chance and the chance any price needs.