Expected Value Calculator

Expected Value (EV) is the single most important concept in gambling mathematics. It tells you exactly how much you can expect to win or lose on average per bet. This calculator reveals the mathematical truth behind any wager, showing why casinos always win in the long run and why some bets are worse than others.

Common Casino Bets

Click any bet to see its expected value:

Understanding Expected Value

Expected value is the cornerstone of gambling mathematics. According to the Encyclopedia Britannica, expected value represents the average outcome you would expect over many repetitions of a random event. In gambling, it tells you exactly how much you can expect to win or lose per bet over the long term.

The formula is deceptively simple:

For a simple win/lose bet like roulette red/black on a European wheel:

• Win probability: 18/37 = 48.65%
• Win amount: +$10 (even money payout)
• Lose probability: 19/37 = 51.35%
• Lose amount: -$10

EV = (0.4865 × $10) + (0.5135 × -$10) = $4.87 - $5.14 = -$0.27

That -$0.27 represents a 2.7% house edge. On every $10 bet, you can expect to lose an average of 27 cents. It doesn't matter how you bet or what patterns you follow—mathematics doesn't care about streaks or intuition.

Why Casinos Always Win

The American Gaming Association reports that commercial casinos generate over $60 billion annually in gross gaming revenue. This isn't luck—it's mathematics working exactly as designed.

Every casino game has a negative expected value for players:

As research from the UNLV International Gaming Institute demonstrates, casinos profit not from any single bet but from the aggregate of millions of bets where negative EV consistently favors the house.

Positive vs. Negative EV

Negative Expected Value (-EV)

Every standard casino bet has negative EV for players. This means:

• You will lose money on average over time
• No betting system can overcome -EV
• The more you bet, the more you lose
• Short-term wins are temporary variance, not skill

Positive Expected Value (+EV)

Positive EV situations are extremely rare in gambling. They occur in:

• Card counting in blackjack: Skilled counters can achieve +0.5% to +2% edge under specific conditions. See our story on the MIT Blackjack Team.
• Specific promotions: Casinos occasionally offer bonuses or match-play coupons that temporarily create +EV situations.
• Poker vs. weaker players: Unlike casino games, poker is player vs. player, so skilled players can achieve +EV against less skilled opponents.
• Sports betting with superior information: Professional bettors with better analysis than oddsmakers can find +EV lines, though the vig makes this difficult.

Understanding when you're in a -EV situation (always, in standard casino play) is essential for making informed decisions about gambling. As noted by the National Council on Problem Gambling, understanding the mathematics of gambling is a key component of responsible gambling education.

The Law of Large Numbers

Expected value works through the Law of Large Numbers, a fundamental theorem proven by mathematician Jacob Bernoulli in 1713. It states that as the number of trials increases, the average outcome approaches the expected value.

This has profound implications for gambling:

• Short term: Anything can happen. Variance is high.
• Medium term: Results start clustering around expected value.
• Long term: Your results will match expected value almost exactly.

Our Session Outcome Calculator shows this in action—short sessions have high variance, but the more bets you make, the more certain your results become. And for negative EV games, that certainty means losses.

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