The tempting wrong answer
On one roll of a fair die the chance of a six is 1/6. A natural thought is that with four rolls the chances add up: 4 x 1/6 = 4/6, which is about 67 percent. The true answer is about 52 percent. The reason is that adding probabilities only works for events that cannot happen together. Getting a six on the first roll and a six on the second roll can both happen, so the “adding” counts those overlaps twice.
The addition error becomes obvious if you push it. With six rolls it would give 6/6, a certainty, and with seven rolls more than 100 percent. But even after a hundred rolls, you cannot be fully certain of a six. Adding probabilities wrongly is one of the most common errors in basic probability.
Use the opposite event
Counting every way to get “at least one six” is awkward: you would need exactly one six, exactly two, and so on. The complement rule gives a shortcut. The opposite of “at least one six” is “no sixes at all”, and the two chances sum to 1.
So: P(at least one six) = 1 - P(no sixes).
On one roll, the chance of not rolling a six is 5/6. Rolls are independent, so for four rolls the chance of no sixes is found by multiplying:
(5/6)^4 = 625 / 1296, which is about 0.482
Therefore
P(at least one six in 4 rolls) = 1 - 625/1296 = 671/1296, which is about 0.518, or 51.8 percent.
Check that 6^4 = 1296 and 5^4 = 625. Both are right, and 1296 - 625 = 671.
Values for other numbers of rolls
The same method gives the chance for any number of rolls n: 1 - (5/6)^n.
| Rolls | Chance of at least one six |
|---|---|
| 1 | 16.7% |
| 2 | 30.6% |
| 3 | 42.1% |
| 4 | 51.8% |
| 6 | 66.5% |
| 10 | 83.8% |
You can check the 2-roll value by hand: 1 - 25/36 = 11/36, which is 30.6 percent. Another way to get 11/36 is to list the grid of 36 outcomes and count those containing a six, as in the two-dice table: 11 cells.
Four is the smallest number of rolls for which the chance goes above one half. After three rolls it is 91/216, about 42 percent, and after four it passes 50 percent. Notice also that 6 rolls does not give certainty: it gives about 66.5 percent, which is close to the figure that the wrong “4/6” method claimed for just four rolls.
Why the answer keeps climbing slowly
Each extra roll multiplies the “no six yet” chance by 5/6. That shrinks it by about 17 percent of itself each time, so it falls steadily but never reaches zero. Because the shrinking is by a fraction, early rolls make the most difference in percentage terms and later ones add less. It will take 13 rolls for the chance to pass 90 percent: (5/6)^13 is about 0.093, leaving about 90.7 percent. At 12 rolls it is about 88.8 percent.
Something to keep clear
This calculation is about the chance of at least one six across a set of rolls viewed in advance. It does not mean a six becomes more likely after several misses. If you have rolled four times without a six, the chance of a six on the fifth roll is still 1/6. The chance of at least one six across five rolls, measured from the start, is 1 - (5/6)^5, about 59.8 percent, but that figure includes the first four rolls, which are already behind you and cannot change. Confusing the two is a form of the gambler’s fallacy, as is the belief in a law of averages that pushes results back toward balance.
Where else the method applies
The same approach works for many “at least one” questions: at least one head in several coin tosses, at least one critical hit in several attacks, or at least one correct guess in several tries. Each time you find the chance of none, using independence, and subtract from 1. It requires events to be independent, with the same probability each time. For many sequential independent tries the binomial distribution generalises it, giving the chance of exactly k successes.
The historical version of this problem, about whether to bet on a six in four rolls, is covered in the article on the de Mere problem.
A word of caution
These are exact probabilities for fair dice, and they say nothing about being able to win money. In a real gambling game, payouts are set so that the operator keeps an edge, and knowing the chance of a six does not remove it. Gambling can be harmful; if it is affecting you, a local support service can help.