How to Compare Two Bets With Different Probabilities and Different Payouts

Most gambling decisions feel like comparisons between apples and oranges. One bet wins 45 percent of the time and pays even money. Another wins 5 percent of the time and pays 15 to 1. Which one is better? The payout on the second looks more exciting, but the win rate on the first looks more reassuring. Without a shared unit of measurement, it is genuinely hard to choose.

Expected value is that shared unit. Once you calculate it for both bets, the comparison becomes straightforward.

What Expected Value Means

Expected value is the average result you would see if you played the same bet an enormous number of times. It does not predict what happens on your next bet. It tells you what one dollar wagered is worth, on average, over the long run.

The calculation has two parts: the gain when you win, and the loss when you lose. You weight each part by its probability and add them together.

The formula, for a simple bet where you risk one dollar:

Expected value = (Probability of winning x Amount won) + (Probability of losing x Amount lost)

Amount lost is expressed as a negative number, because it is money leaving your pocket.

A Step-by-Step Example With Two Hypothetical Bets

Imagine a hypothetical casino game with two available bets. Call them Bet A and Bet B.

Bet A: Wins 45 percent of the time. Pays even money, meaning a one-dollar bet returns one dollar in profit.

Bet B: Wins 5 percent of the time. Pays 15 to 1, meaning a one-dollar bet returns fifteen dollars in profit.

To find the expected value of each bet, first convert the percentages to decimals. 45 percent becomes 0.45. 5 percent becomes 0.05. The probability of losing is whatever is left: 0.55 for Bet A and 0.95 for Bet B.

Bet A expected value: (0.45 x $1.00) + (0.55 x -$1.00) = $0.45 - $0.55 = -$0.10

For every dollar wagered on Bet A, the expected average loss is ten cents.

Bet B expected value: (0.05 x $15.00) + (0.95 x -$1.00) = $0.75 - $0.95 = -$0.20

For every dollar wagered on Bet B, the expected average loss is twenty cents.

Bet A has a lower expected loss per dollar. On this measure alone, Bet A is the better bet. The larger payout on Bet B is not large enough to compensate for how rarely it lands.

Why Payouts and Probabilities Must Always Be Read Together

This is the core insight. A big payout number only matters relative to how often you win. A 15 to 1 payout sounds generous until you realize the win rate is only 5 percent. The math makes the comparison concrete rather than impressionistic.

You can also express expected value as a percentage of each dollar wagered, which is directly comparable to the house edge figure you might see published for a game. Multiply the expected value by 100. Bet A costs 10 percent per dollar wagered. Bet B costs 20 percent. Bet B is twice as expensive per dollar, even though it has the bigger payout.

One Important Limit of Expected Value

Expected value describes what happens on average over a very large number of bets. Over a short session, your actual results will swing well above and below that average. This is called variance, and it matters especially for bets like Bet B.

Bet B wins only 5 percent of the time. That means, on average, you would expect to go roughly 20 bets without a win. In a short session, you might go much longer without a win, or you might win twice in five bets. Either is plausible. Bet A, which wins nearly half the time, will produce a session experience that feels closer to the expected value more quickly, simply because wins and losses alternate more frequently.

So two things are true at once. Expected value tells you the long-run cost per dollar. Variance tells you how wild the ride will be getting there. A bet with a worse expected value but lower variance might actually be preferable depending on your session length and bankroll size. Expected value is the right tool for comparing the cost, but it is not the only number worth understanding.

Applying This to Real Decisions

When you encounter any bet, you can run this calculation yourself as long as you know two things: the probability of winning and the payout. Both should be available from a game's published rules or paytable.

Start by writing down the probability of winning as a decimal. Subtract it from 1.00 to get the probability of losing. Multiply winning probability by the profit per dollar wagered. Multiply losing probability by -1.00 (or whatever you would lose). Add the two results. If the number is negative, the bet has a built-in cost, which is the case for every standard casino game. The closer that number is to zero, the lower the long-run cost.

If you want to compare two bets directly, simply run the calculation for each and look at the results side by side. The bet with the smaller negative number costs you less per dollar wagered.

A Note on Assumptions

This framework assumes the probabilities you are using are accurate. Published house edge figures for regulated games are generally reliable, but any bet with an unverified probability estimate introduces uncertainty into the calculation. The math is only as trustworthy as the inputs.

It also assumes the game's probabilities are fixed and do not change based on previous outcomes, which is true for most casino games and most standard bets. Games where earlier results change the remaining probabilities, such as card games played from a finite deck without reshuffling, require a more involved calculation.

For the majority of bets you will encounter, though, this single calculation gives you a clear, honest comparison that no amount of intuition about exciting payouts or frequent wins can replace.