Expected value (definition and examples)

Expected value (EV) is the average amount a bet returns per round if you could repeat it endlessly. You find it by multiplying each possible result by its chance and adding the products. It is the number behind RTP and house edge.

An EV below your stake means the bet loses on average; above, it gains on average. Casino-style games are built so the EV is slightly below the stake.

Worked example: two roulette bets

Cashout Arcade's Roulette has 37 pockets (0 to 36). Every bet is priced to return 97 percent, but the shape differs. Take a 100-coin bet.

A straight number. It wins in 1 pocket out of 37 and pays 35.89x, including your stake, so a win returns 3,589 coins.

EV = (1 / 37) x 3,589 + (36 / 37) x 0 = 97.0 coins

Red. It wins in 18 pockets out of 37 and pays 1.9938x, so a win returns 199.38 coins.

EV = (18 / 37) x 199.38 + (19 / 37) x 0 = about 97.0 coins

(The multiplier is rounded down at the fourth decimal, which puts the red bet a fraction of a coin under 97.)

Both bets have the same EV, so neither is a "better value" bet. They differ in volatility: the number wins about 1 time in 37 for a big amount, red wins nearly half the time for a small one. This is why EV and swing are two separate ideas.

What EV does and does not tell you

EV is a statement about the long run. One spin of the red bet returns either 0 or 199.38, never 97. Over 10 spins the result is mostly luck. Only over thousands of rounds does the average approach the EV, as explained in Law of large numbers (casino example).

Common misconceptions

• "Positive EV in one round means I will win." A single round is still random.
• "I can raise EV by changing bet size or timing." In games with independent rounds, EV is a percentage of the wager. A bigger bet scales the expected loss with it.
• "A big top prize raises EV." The payout table is balanced so that every game, big top prize or not, lands near the same 97 percent.

For skill games such as Blackjack, EV depends on your decisions; see Basic strategy (blackjack definition). Every game's return is on the Odds page. Coins are virtual with no cash value, so EV here describes game balance, not income.