Expected Value Calculator

Understanding Expected Value

Expected Value (EV) is one of the most fundamental concepts in probability theory and gambling mathematics. It answers a simple but crucial question: "If I make this bet repeatedly, what will happen to my money on average?"

The Formula

Expected value is calculated by multiplying each possible outcome by its probability, then adding them together:

Example: Fair Coin Flip

Imagine a coin flip where you win $1 on heads and lose $1 on tails. Each outcome has a 50% (0.5) probability:

An EV of $0 means this is a fair game—you neither gain nor lose money on average.

Example: Roulette Single Number

In roulette, betting on a single number pays 35:1, but there are 37 numbers (including zero) on a European wheel. With a $1 bet:

• Probability of winning: 1/37 = 2.7027%
  Payout if you win: $35 (you get your $1 back plus $35 profit)
• Probability of losing: 36/37 = 97.2973%
  Payout if you lose: -$1

This negative EV of -$0.027 (or -2.7%) means that for every dollar you bet on a single number in roulette, you lose about 2.7 cents on average. This is the house edge.

Why It Matters

Every casino game, lottery, and sports bet has an expected value. Understanding EV helps you:

• Recognize which bets lose money faster (higher house edge = more negative EV)
• Understand why "the house always wins" in the long run
• Make informed decisions about where to place your entertainment budget if you choose to gamble
• Appreciate the mathematical impossibility of long-term profit from negative-EV games

Professional gamblers and investors use expected value to identify opportunities where the odds are in their favor (positive EV). In casino gambling, however, virtually every bet has negative EV—which is exactly how casinos stay profitable.

Common Misconceptions

• "I won last time, so the EV must be positive." Individual results don't change the mathematical expectation. You can win many times with negative EV; the law of large numbers only applies over thousands of trials.
• "If I double my bet after a loss, I'll come out ahead." Betting systems like the Martingale don't change the expected value. They only redistribute when wins and losses occur, often increasing risk dramatically.
• "This slot machine hasn't paid out in a while, so it's due." Each spin is independent. The expected value remains constant regardless of past results.