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Every board game, every cricket toss and every “heads or tails” argument rests on the same idea: some outcomes are equally likely, so counting them gives you the odds. Counting is easy. What goes wrong is intuition, which finds patterns in streaks and believes that luck must “even out” in the next roll.
One die: simple, and boring on purpose
A fair die has six faces, each with the same chance of landing up: 1 in 6, or about 16.7%. That means each number is “due” no more often than any other. Add a second die and things become interesting, because the total is no longer equally likely.
Two dice: counting the 36 outcomes
Treat the two dice as different (say one red, one blue). There are 6 × 6 = 36 equally likely pairs. The table below shows the total for every pair. Look along the diagonals: each diagonal has the same sum, and the middle one is the longest.
| Red die | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Columns show the blue die. Darker means the total is more likely. The ringed cell is a 1 and a 6, which makes 7.
The totals form a triangle: 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1. Seven has six ways and is the most likely total.
A total of 7 comes up about one roll in six. A total of 12 (double six) comes up about one roll in thirty-six.
The middle of the range collects most of the outcomes, which is why 6, 7 and 8 come up so often in dice games.
A gambler’s puzzle from the 1600s
A French nobleman, the Chevalier de Méré, noticed that he made money betting on at least one six in four rolls of a die, but lost on at least one double six in 24 rolls of two dice. The two bets sound similar, since 4 is to 6 as 24 is to 36. The reason they are different is that “at least one” is easiest to count through its opposite: no success at all.
- 1Chance of no six on one roll5 out of 6, which is about 0.833.
- 2Chance of no six in all four rolls(5/6)⁴ = 0.482, because the rolls are independent.
- 3Flip it around1 − 0.482 = 0.518, so a six turns up 51.8% of the time.
- Win: a six in 4 rolls51.8%
- Lose48.2%
A small edge for the player, enough to win steadily over many games.
- Win: a double six in 24 rolls49.1%
- Lose50.9%
Barely under half. Chance of no double six per roll is 35/36, and (35/36)²⁴ = 0.509. Pascal and Fermat solved this around 1654, which helped start probability theory.
The coin has no memory
Suppose a fair coin lands heads five times in a row. Many people feel tails is now “due”. It is not. Each flip is independent: the coin does not know what happened before. The chance of five heads in a row before you start is small, but the chance of a sixth head after five is just 50%.
Each extra head halves the chance. Five in a row is 3.1%, ten in a row is about 0.1%. Rare, but across many flips you will see them.
What really settles down
A coin was flipped by computer in an experiment, 5,000 times with a fixed seed so the result is repeatable. The share of heads wandered at first, then calmed down:
This is the law of large numbers: the share of heads gets closer to 50% as flips grow.
The common mistake is to think the count of heads and tails must also even out. It does not have to. A lead of 40 extra heads after 1,000 flips is 52% heads, but after 100,000 flips it is only 50.02%, even though the gap is exactly the same. Chance does not correct the past; it simply buries it under new flips.
Quick practice
- What is the chance of a total of 9 on two dice? 4 of 36, which is 11.1%.
- Which two totals are equally likely as 5? 9, since both have 4 ways.
- After four tails in a row, what is the chance the next flip is heads? 50%.
- What is the chance of at least one head in 3 flips? 1 − (1/2)³ = 87.5%.
Randomness has a shape, and counting reveals it. For where randomness meets security, read about UUID collisions and password entropy.
Questions people ask
What is the most likely total with two dice?
Seven. It can happen in 6 of the 36 ways, which is 16.7%.
What is the chance of rolling a double six?
1 in 36, or about 2.8%.
Is a coin more likely to land tails after many heads?
No. Each flip is independent. A fair coin is always 50:50, no matter what came before.
What is the chance of rolling at least one six in four rolls?
About 51.8%. The chance of no six at all is (5/6)⁴ = 48.2%, so at least one six is the rest.
What is the law of large numbers?
As the number of trials grows, the average result (such as the share of heads) approaches the true probability. It does not mean the raw counts must even out.
Are online dice rollers truly random?
They use a pseudo-random number generator, which is fine for games. Anything that needs true unpredictability, such as passwords, should use a cryptographically secure source.