What expected value means
Expected value is the average result you would get if you could repeat the same random situation a very large number of times. It is a weighted average: each possible outcome is multiplied by its probability, and the products are added together.
The word “expected” is slightly misleading. It does not mean the result you should anticipate on any single try. A fair six-sided die has an expected value of 3.5, yet it can never roll a 3.5. The number describes the centre of the long-run picture, not a single event.
The basic calculation
The recipe has three steps.
- List every possible outcome and its value (points, damage, coins, or whatever you care about).
- Find the probability of each outcome. The probabilities should add up to 1.
- Multiply each value by its probability and add up the results.
For a fair die, each face has probability 1/6, so the expected value is (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21 / 6 = 3.5.
This works only when you have the probabilities right. Listing the full sample space first, and checking that no outcome is missed or counted twice, prevents most arithmetic slips.
A worked example with a stake
Suppose a game asks you to pay 2 coins to roll one die. If it shows a six, you receive 9 coins. Otherwise you receive nothing.
- Chance of a six: 1/6. Payout: 9 coins.
- Chance of anything else: 5/6. Payout: 0 coins.
The expected payout is (1/6) × 9 + (5/6) × 0 = 1.5 coins. You paid 2, so the expected net result is 1.5 − 2 = −0.5 coins per play. On average, this game costs half a coin each time.
The fair price for this roll would be 1.5 coins, the point where the expected net result is exactly zero. Anything below that favours the player on average, and anything above favours the game.
Comparing two choices
Expected value is most useful when you must pick between options. Imagine a turn where you can try one of three moves, each scoring points only if it succeeds:
| Move | Chance of success | Points if it works | Expected points |
|---|---|---|---|
| A | 70% | 4 | 0.70 × 4 = 2.8 |
| B | 40% | 7 | 0.40 × 7 = 2.8 |
| C | 25% | 12 | 0.25 × 12 = 3.0 |
Moves A and B are tied on average, and C is slightly ahead. Notice what the table does not say: A succeeds most often and C fails three times out of four. Which one you prefer may depend on whether you need steady progress or a big swing, which is where spread, not just the average, starts to matter.
Reading the sign of the result
The sign of the expected net result gives a quick verdict. Positive means that, over many repeats, the choice tends to leave you ahead. Negative means it tends to leave you behind. Zero means the situation is fair in the long run. The size matters too: an expected net of −0.5 on a 2-coin stake is a 25% average cost per play, which is steep, while −0.01 on a 1-coin stake is about 1%.
Be careful to measure everything that changes hands. A fair-looking payout table can hide a cost in the entry fee, a rule that voids some wins, or a resource you spend on each attempt. Include every cost and every payout, in the same units, before judging the result.
Value is not always money
Expected value works with any numerical measure, not only coins. In a board game you might count points, cards drawn or tiles gained. In a video game you might count damage, experience or progress toward a goal. The only requirement is that the quantity is something you actually want more of, and that you can assign it a number.
When outcomes are not naturally numeric, you can assign scores yourself, such as +1 for winning the exchange and −1 for losing it. These scores are judgements, so a different but reasonable scoring can change the ranking of your options. It is worth testing whether your conclusion survives a few alternatives.
Adding up expected values
Expected values add cleanly, even when the events are connected. The expected total of two dice is 3.5 + 3.5 = 7. A turn with three independent attacks has an expected total equal to the sum of each attack’s expected value. This makes large problems manageable: break them into small pieces, find each piece’s average, then add.
Multiplying probabilities, by contrast, is for the chance of several things all happening, and it follows the multiplication rule for sequential events. Keep the two ideas separate: you add expected values, and you multiply probabilities of chained events.
Common slips
- Probabilities that do not add to 1. If your list of outcomes sums to more or less than 100%, something is missing or double counted.
- Using the payout instead of the profit. If a prize of 9 includes the returned stake, subtract what you paid accordingly.
- Forgetting the zero outcomes. Losing outcomes belong in the sum even when they contribute nothing.
- Treating the average as a promise. Expected value describes many repeats, so a single result can sit far away from it.
Where the average stops helping
Expected value answers one question: what happens on average over many repeats. It says nothing about how far a short run can wander from that average. A game with a small positive expected value can still lose resources for a long stretch, and a game with a negative one can pay out in the short term. That is the reason short sessions can look nothing like the averages.
Spread matters too. Two choices with the same average can feel completely different, and a very lopsided payout can make the average describe almost nobody’s actual experience.
Expected value also leaves out things you may care about, such as the fun of a gamble, the need to avoid a bad outcome you cannot recover from, or the way people tend to weigh losses more heavily than equal gains, a pattern known as loss aversion.
Using it responsibly
In casino-style games, the house edge is simply a negative expected value built into the rules. Understanding it explains why a game costs money over time, not how to avoid that cost. No method of choosing bets changes the average of a game whose rules are fixed, and results in any given session are uncertain.
In tabletop and video games, the same arithmetic is a practical tool for informed decisions: comparing attacks, pricing items or judging whether a risky play is worth it. Treat the number as one input, along with the stakes, the alternatives and how much variation you can tolerate. When you can estimate it, the expected value gives you a clear, honest baseline, and that is most of what it is for.