Expected Value in Games

Discrete Math · Axiom Academy

REAL WORLD Expected Value in Games Discover the mathematics behind casinos and lotteries—and why the house always wins in the long run. Walk into any casino, and you'll see flashing lights, excited players, and occasional big winners. But here's the mathematical truth: casinos are guaranteed to profit —not through luck, but through expected value. Expected value (EV) is the average outcome you can expect over many repetitions of a random event. It's calculated by multiplying each possible outcome by its probability, then summing them all together. Let's explore this concept by analyzing real casino games and discovering the house edge —the built-in advantage that ensures casinos profit long-term. American roulette has 38 spaces: numbers 1-36 (half red, half black), plus 0 and 00 (both green). Let's analyze the most common bet: betting 10 on red . Now let's calculate the expected value of a 10 bet on red. Remember, you either: The house edge is the percentage of each bet that the casino expects to keep long-term. It's calculated from the expected value: ✓ For every 100 wagered on roulette, the casino expects to keep 5.26 ✓ Individual players may win or lose, but across thousands of bets, the casino's profit is mathematically guaranteed ✓ This isn't cheating—it's probability and expected value at work! Different casino games have different house edges. Let's compare popular games to see which ones favor the house most: The Lottery: Extreme Negative Expected Value

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