Expected Value of a Bet

Expected value weights each outcome by its probability: win W a fraction p of the time, lose L the rest. Bet $1 on a single number at American roulette and you win $35 with probability 1/38 and lose $1 with probability 37/38, so E = (1/38)(35) − (37/38)(1) = −2/38 ≈ −$0.0526 — the famous 5.26% house edge, identical on every bet the American wheel offers except the five-number line. Enter p as a decimal from 0 to 1, or use the % unit; W and L are plain amounts in whatever currency you like.

Rearranged for p, it gives the break-even chance a wager needs to be worth taking: risking $1 to win $4 requires p = (0 + 1)/(4 + 1) = 0.2, so anything above a 20% chance is profitable in the long run. That framing — comparing your estimated probability against the price on offer — is how professional bettors and insurers actually work. The trap is treating expected value as what will happen: you never lose 5.26 cents on a spin, you lose $1 or gain $35, and it is only across thousands of spins that the average asserts itself. Expected value also ignores risk of ruin, which is why the St Petersburg game, with infinite expectation, is worth only a few dollars to a real player.