Expected value (EV) is the average result you would get if you repeated a roll or a bet a very large number of times. It is not what will happen on any single throw — it is the long-run centre of gravity that every outcome pulls toward. Master this one formula and the fog around dice odds lifts.
The expected value of a single die
For a fair die, every face is equally likely, so the average is simply the midpoint of its faces. A die with n sides has an expected value of:
EV = (n + 1) ÷ 2
That gives clean, memorable numbers for the standard polyhedral set:
| Die | Faces | Expected value |
|---|---|---|
| d4 | 1–4 | 2.5 |
| d6 | 1–6 | 3.5 |
| d8 | 1–8 | 4.5 |
| d10 | 1–10 | 5.5 |
| d12 | 1–12 | 6.5 |
| d20 | 1–20 | 10.5 |
The expected value of a sum
Here is the property that makes EV so powerful: expected values add. The average of a pool of dice is just the sum of each die’s average, and a flat modifier simply shifts it. You never have to enumerate all the combinations to find the mean.
- 2d6 = 3.5 + 3.5 = 7 — which is why 7 is the heart of any two-dice game.
- 3d6 = 3 × 3.5 = 10.5.
- 4d6 = 4 × 3.5 = 14.
- 1d20 + 5 = 10.5 + 5 = 15.5.
The averages add cleanly even though the shape of the results does not — a single d20 is flat, while 2d6 bunches around 7. EV tells you the centre; it does not tell you the spread. (For the spread, see probability, variance and luck.)
The expected value of a bet
For a wager, multiply each outcome’s payoff by its probability and add them up. Take a simple even-money bet on rolling a 4, 5 or 6 on one d6: you win 1 unit half the time and lose 1 unit half the time.
EV = (0.5 × +1) + (0.5 × −1) = 0
Zero EV means a fair bet — break-even in the long run. Now make it realistic. Suppose the same bet only pays if you roll a 5 or 6 (chance 1/3) but still pays even money:
EV = (1/3 × +1) + (2/3 × −1) = −1/3 ≈ −0.33
You lose about 33p per pound over time. That negative number is the house edge. In fact, house edge is nothing more than expected value made negative and expressed as a percentage of your stake. When a casino game quotes a “1.4% edge”, it is saying the EV of the bet is −0.014 per unit. Every dice bet ever devised can be judged this way — a table of chances times payoffs, summed.
Try building the intuition live: roll a die a few hundred times in Dice Drop and watch the running average settle toward its expected value even while individual rolls swing wildly.
Using EV to decide: press or bank?
EV is not only for judging casino bets — it is the tool for “push your luck” games like Farkle, Pig or Cee-lo, where you choose whether to bank points or roll again for more. The rule is simple: roll again only when the expected gain outweighs the expected loss.
Compare two quantities before each decision:
- What you risk — the points you have banked this turn, multiplied by your chance of busting.
- What you stand to gain — the average points a successful roll adds, multiplied by your chance of surviving.
When your pile is small, the risk of busting costs little, so pressing on is usually +EV. As your banked points grow, the same bust probability threatens a bigger loss, and eventually the maths flips negative — that is your cue to bank. The precise tipping point depends on the game’s scoring, but the principle is universal: keep rolling while the expected value of continuing is positive, and stop the moment it turns.
The same lens explains why casino games feel “fun but losing”: they are engineered so the EV of every available bet is slightly negative. You cannot out-strategy a negative EV — you can only pick the bets closest to zero and manage how long you play.
Try it yourself. Dice Drop rolls real physics dice in your browser — advantage, keep-highest and modifiers built in.
🎲 Roll dice freeKeep reading: the maths of keep-highest and keep-lowest — how dropping a die shifts the average away from the simple sum.