Expected Value in Sports Betting, Explained
Related book: The Sports Investing Blueprint by Ethan Cole · Amazon
Expected value tells you how to spot a +EV price: if the chance you assign is above the break-even rate of the odds, the price is plus-EV. Line shopping is how you find a better number — and a better number is how you improve that average.
The formula and the example tables below show the arithmetic. For the break-even rate of a single price, see implied probability and the vig. A longer list of American prices is in the American odds to implied probability table.
The formula
Pick a stake. If the bet wins, the profit is what you are paid besides the stake being returned. If the bet loses, the loss is the stake.
EV = (chance of winning × profit if win) − (chance of losing × stake)
The two chances must add to 1. For a two-way bet with no other result, the chance of losing is 1 minus the chance of winning.
The profit at an American price comes from the decimal form of that price. For a negative price, drop the minus and call that number A. The decimal is (A + 100) / A. For a stake S, the profit if the bet wins is S × (100 / A). For -110, A is 110, the exact decimal is 21 / 11, and the profit on a $100 stake is $100 × (100 / 110) = 1,000 / 11, which is 90.9090…, displayed $90.91.
Example: -110 at a 50% chance
Use a $100 stake at -110, and assign a 50% chance of winning.
The chance of losing is also 50%. The exact profit if the bet wins is 1,000 / 11.
EV = (0.50 × 1,000 / 11) − (0.50 × 100) = 500 / 11 − 50 = −50 / 11
−50 / 11 is −4.5454…, displayed −$4.55.
| Input | Exact | Displayed |
|---|---|---|
| Price | -110 | -110 |
| Stake | $100 | $100 |
| Profit if win | 1,000 / 11 | $90.91 |
| Assigned win chance | 1 / 2 | 50% |
| Expected value | −50 / 11 | −$4.55 |
The average is negative. That is the same fact as the break-even rate on the vig page: -110 implies 11 / 21, which is 52.38%. A 50% chance sits below that rate, so this price is minus-EV at that chance. The −$4.55 is the average over many identical trials. The next single bet still returns either +$90.91 or −$100.
Example: the same price at a 55% chance
Keep the $100 stake at -110, and assign a 55% chance of winning, which is 11 / 20. The chance of losing is 45%, or 9 / 20.
EV = (11 / 20 × 1,000 / 11) − (9 / 20 × 100) = 1,000 / 20 − 45 = 50 − 45 = 5
The displayed figure is +$5.00. It is exact in this setup.
The average is positive because the assigned chance is above the 52.38% break-even rate of -110. The input has to be the chance you actually believe. If the true chance were still 50%, the first table would be the one that applies.
A better price changes the average
Return to the 50% assignment and the $100 stake, and compare -110 with -105. For -105, A is 105. The exact decimal is 205 / 105, displayed 1.95. The profit if the bet wins is $100 × (100 / 105) = 10,000 / 105 = 2,000 / 21, which is 95.2380…, displayed $95.24.
EV at -105 = (0.50 × 2,000 / 21) − (0.50 × 100) = 1,000 / 21 − 50 = −50 / 21
−50 / 21 is −2.3809…, displayed −$2.38.
| Price | Profit on $100 if win | EV at a 50% chance |
|---|---|---|
| -110 | $90.91 | −$4.55 |
| -105 | $95.24 | −$2.38 |
The better price improves the average by $2.17 on this $100 stake. At a 50% chance the average is still negative, because the break-even rate at -105 is 105 / 205, which is 51.22%. Line shopping is the habit that compares legal prices before you take one, so you can capture a move like this.
A positive average can still lose
If the 55% assignment in the second example were the true chance, the average would be +$5.00 on $100, and 45 of every 100 identical bets would still lose. A short run of losses would not disprove the average. A short run of wins would not prove it. Understanding variance is that limit in results: a handful of bets is a noisy grade of any long-run figure.
The same limit runs the other way. A winning bet at a negative expected value is still a bet whose average, at the chance you used, was a loss. The result of one trial does not rewrite the arithmetic.
EV and stake size
Bankroll management and unit sizing asks how large the stake is next to a separate bankroll. A +$5.00 average on $100 does not, by itself, tell you whether $100 is a small fraction of that bankroll.
The bankroll and line-shopping worksheet holds single-price examples: a flat stake, a stake-to-win, and a fixed or resized streak. You can use it to record the stake you chose.
Bottom line
Expected value is (chance of winning × profit if win) minus (chance of losing × stake). At -110 and a $100 stake, a 50% assigned chance yields −$4.55, and a 55% assigned chance yields +$5.00. The same 50% chance at -105 yields −$2.38. A better price improves the average. Line shopping is how you look for that price.
For how money management sits next to tools, systems, and scam avoidance, see the book.