Sports betting EV calculator: the expected value of any bet

Expected value (EV) is what a bet is worth on average if you could place it over and over: the profit you win times the chance you win it, minus the stake you lose times the chance you lose it. A positive number means the price is better than the odds of winning justify. It doesn't mean the bet wins.

The formula is short. Getting a trustworthy probability to put into it is the hard part, and most of this page is about that.

The formula

EV = (probability of winning x profit if it wins) - (probability of losing x stake)

Profit comes from the price. At -110, a 100 stake wins 90.91. At +120, it wins 120. At +100, it wins 100.

Example 1: a fair coin at -110

Both sides have a 50 percent chance. A 100 bet at -110:

EV = 0.50 x 90.91 - 0.50 x 100 = -4.55

On average this bet loses 4.55 per 100. That's the vig, and it's why betting two even sides at -110 has a negative expectation.

Example 2: the same bet, a better price

Suppose the true chance is 52 percent and the price is +100:

EV = 0.52 x 100 - 0.48 x 100 = +4.00

Every 100 staked is worth 4 on average. The break-even probability at +100 is 50 percent, so 52 percent clears it by two points. At -110, break-even is 110 / 210 = 52.38 percent, so a 52 percent estimate at -110 is slightly negative.

Break-even probability = the implied probability of the price. Any bet is +EV exactly when your probability is higher than that number.

Where the probability comes from

Your estimate of the winning chance is the input that decides everything. There are three common sources, in rising order of trust:

1. Your own model or opinion. Easy to make, hard to check, and easy to overrate.
2. A consensus of books. Better, but you are measuring an average that includes books with wide margins.
3. A de-vigged price from sharp books. The margin is removed from the books that take the most informed money. This is the reference quant+ uses, and the probabilities implied by public odds are information, enough that researchers have used them to look for mispriced bets (arXiv, 2017). The method, and how to calculate it by hand, is in the fair odds calculator guide.

Take a game with posted prices that devig to Team A at 54.27 percent. Another book offers A at -110 (the margin-free price is a good reference because betting odds are frequently found to outperform mathematical models in sports forecasting (PLOS ONE, 2018)):

EV per 100 = 0.5427 x 90.91 - 0.4573 x 100 = +3.61

That figure measures the gap between one book's price and the market's view with the margin taken out. It says the price is better than the market's own estimate. It doesn't say the market's estimate is right, only that you are comparing against the best reference available.

EV for parlays

For a parlay, multiply the leg probabilities and use the combined payout. Two legs at 54.27 percent each, assumed independent, give 29.45 percent. A book paying +260 on the pair:

EV per 100 = 0.2945 x 260 - 0.7055 x 100 = +6.02

The independence assumption does the work here. Legs from the same game are often related, so the true joint probability can be higher or lower than the product. If you overestimate it, the parlay looks +EV when it isn't. Parlays also stack the margin from every leg, which is why a parlay that looks attractive often turns out to be priced against you.

Positive EV doesn't mean you'll win

EV is an average over many bets. A single bet, or a hundred, can go the other way.

Take that +4.00 per 100 example on even-money bets with a 52 percent win chance. Over 100 bets, the expected profit is +400. The standard deviation of the total is about 1000 (each bet swings about 100 either way, and the swing grows with the square root of the number of bets). That puts 0 only 0.4 standard deviations below the mean, so roughly a third of 100-bet samples at this edge would still finish below zero. These are our arithmetic and an illustration, not results.

So a hot or cold streak over a hundred bets tells you very little about whether the edge is real. The closing line is a faster check: it measures whether your price beat where the market ended up, without waiting for outcomes to average out.

Sizing: Kelly and why fractions

If you do have a real edge, how much to stake is a separate question. The Kelly criterion picks the stake that maximizes the expected value of the logarithm of wealth, which Edward Thorp's paper describes as the growth-optimal strategy. For an even-money bet at 52 percent, full Kelly is 4 percent of bankroll: (b x p - q) / b = (1 x 0.52 - 0.48) / 1.

Full Kelly assumes your probability is exactly right. It never is, and an overestimate turns an apparent edge into a loss. That's why many bettors scale it down. The quant+ scanner's Kelly calculator defaults to a quarter of full Kelly, with half, tenth and full as options, and shows the full-Kelly percentage next to the suggested stake so you can see how much you've scaled it down. It's a sizing aid, not advice. Set your own limit and stay inside it.

How quant+ uses EV

On every board row, the edge column shows the EV of the best available price against the fair price. The positive EV scanner sorts the whole board by it, and the odds comparison view shows which book carries the price. In the bet journal, each bet records the fair probability at the moment you logged it, so you can later compare what the market said then against how it closed.

There is a second way to use it: EV helps you reject bets. A price with no edge over the fair line, or a negative one, is easier to skip when the number is in front of you.

Limits

• The result is only as good as the probability. Garbage in, garbage out. A fair price from sharp books is a reference, not the truth.
• EV ignores variance, limits and fees. A +2 percent edge may not survive a stake limit, a withdrawal fee or a stale line.
• Closing-line research has its own caveat. A study of NFL money lines found that inefficiencies exist but are unpredictable and vary by season, so an edge that showed up in one season may not repeat in the next.
• quant+ does not place bets or tell you what to bet. It shows the price, the fair line and the edge.

Quant+ is analytics for informational purposes. 21+. If you or someone you know has a gambling problem, call 1-800-GAMBLER.