What expected value measures
Expected value is the average result of a bet if it were repeated a very large number of times. To find it, list every possible outcome, multiply each by its probability, and add the products. It does not say what will happen on the next hand. It describes where the average settles.
For a bet where you can win or lose, the average result per unit bet is the sum of the possible gains weighted by their chances. A positive figure would mean an average profit. A negative one means an average loss. In blackjack the figure for the player is negative in ordinary play.
A simple illustration
Real blackjack has many outcomes: wins, losses, pushes, naturals, doubles and splits. To see the idea cleanly, start with a simplified bet with round, made-up probabilities, used only to illustrate the arithmetic.
Suppose you bet $10 on a hand, and for illustration:
- you win $10 with probability 0.48
- you lose $10 with probability 0.52
Then:
- 0.48 × (+$10) = +$4.80
- 0.52 × (−$10) = −$5.20
- Expected value = +$4.80 − $5.20 = −$0.40
So the average result is a loss of $0.40 per $10 bet, or 4% of the amount bet. The probabilities add to 1 (0.48 + 0.52), which is a useful check.
Now add a push, where the bet is returned. Suppose win 0.43, lose 0.48, push 0.09. The probabilities add to 1.00. Then:
- 0.43 × (+$10) = +$4.30
- 0.48 × (−$10) = −$4.80
- 0.09 × $0 = $0
- Expected value = −$0.50
The push contributes nothing to the average, but it takes up probability that would otherwise go to wins or losses. These numbers are not real blackjack probabilities. They exist only to practise the method.
Extra payouts change the sum
A natural pays more than even money. Suppose, again for illustration only, that a natural happens with probability 0.048 and pays 3 to 2. On a $10 bet the winning amount is $15, and the contribution to the average is 0.048 × $15 = $0.72. If the same natural paid 6 to 5, the winning amount would be $12, and the contribution would be 0.048 × $12 = $0.576. The difference per $10 bet is about $0.14. That is why the way naturals pay is such a significant rule, and it matches the point made in the discussion of natural payouts.
Why a negative expectation means losses over time
Suppose the expected value is −$0.40 per $10 bet, as in the first example. Over 100 such bets, the average total result is 100 × (−$0.40) = −$40. Over 1,000 bets it is −$400. The average grows with the number of bets because every bet adds the same negative amount.
The key word is average. In real play, the actual result varies a lot around that average. A player could be ahead after 100 bets, or far behind. The spread of results is described by standard deviation, and it grows more slowly than the total of the expected losses. This is why, over a long enough run, the negative expectation tends to show through, a point explained under the law of large numbers. For a small number of bets, the luck of the draw can easily be larger than the expected loss.
It also explains why splitting or changing the pattern of bets does not help. If every bet has a negative expected value, adding bets together only adds negative numbers. Changing the amounts changes the size of the swings, not the sign of the average.
Connecting expected value to the house edge
The house edge is the expected loss expressed as a percentage of the amount bet. In the first example, −$0.40 on a $10 bet is −4%, so the edge would be 4%. Real blackjack edges are usually much smaller, but the idea is identical. The edge is the negative expected value per unit wagered, seen from the casino’s side.
This is also how basic strategy is derived. For each hand and dealer upcard, the expected value of each action is compared, and the action with the best (least negative) value is chosen. The chart is a list of those comparisons.
Common errors
Treating the average as a promise. An expected loss of $0.40 per bet does not mean you lose $0.40 each time. Each bet is a win, loss or push.
Believing a streak changes the expectation. The expected value of the next hand does not depend on whether the last few were wins or losses. Believing otherwise is the gambler’s fallacy.
Ignoring pushes and special payouts. Leaving out ties, naturals, doubles or splits gives a wrong total. In any calculation, make sure the probabilities add to 1.
Using the figures as real odds. The illustration numbers here are not blackjack’s actual probabilities. Real values depend on the rules, the number of decks and the player’s decisions.
What to take away
Expected value turns a bet into one number: the average result per unit staked. In blackjack that number is negative for the player under ordinary conditions, so the sensible way to view play is as paying a price for entertainment. Decide in advance how much you are willing to spend, never bet more than you can afford to lose, and do not chase losses. If gambling is causing harm, contact a local support service.