# Two dice and a coin: why seven wins, and why a streak proves nothing

> A fair die or coin is the simplest random machine there is, yet it fools people every day. This guide uses all 36 outcomes of two dice, the famous 17th-century gambling puzzle and a 5,000-flip experiment to show what chance really does.

Published 2026-10-08 by Toolcloak. 4 minute read. Original: https://toolcloak.pages.dev/articles/two-dice-and-a-coin-why-seven-wins-and-why-streaks-prove-nothing
Related free tool: [Dice Roller & Coin Flip](https://toolcloak.pages.dev/tools/dice-roller-coin-flip)

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.

> **In short**
> - With two dice there are **36 equally likely outcomes**. A total of **7 can happen 6 ways (16.7%)**, while 2 and 12 happen only 1 way each (2.8%).
> - Rolling at least one six in 4 throws of one die is **51.8%**. At least one double-six in 24 throws of two dice is **49.1%**: slightly worse, not better.
> - A fair coin has **no memory**. After five heads in a row the next flip is still 50:50.
> - Over thousands of flips the **share of heads** settles near 50%. The **difference in count** does not shrink.

## 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.

Try it: [Dice Roller & Coin Flip](https://toolcloak.pages.dev/tools/dice-roller-coin-flip). Roll one or several dice or flip a coin. Roll a few hundred times and watch the totals build up the shape described below.

## 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.

**Total of two dice, for every combination**

| 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 |

**How many of the 36 outcomes give each total**

|  | Value |
| --- | --- |
| 2 | 1 |
| 3 | 2 |
| 4 | 3 |
| 5 | 4 |
| 6 | 5 |
| 7 | 6 |
| 8 | 5 |
| 9 | 4 |
| 10 | 3 |
| 11 | 2 |
| 12 | 1 |

*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.*

**Chance of each total (%)**

|  | Chance |
| --- | --- |
| Total 2 | 2.8% |
| Total 3 | 5.6% |
| Total 4 | 8.3% |
| Total 5 | 11.1% |
| Total 6 | 13.9% |
| Total 7 | 16.7% |
| Total 8 | 13.9% |
| Total 9 | 11.1% |
| Total 10 | 8.3% |
| Total 11 | 5.6% |
| Total 12 | 2.8% |

*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.*

**Where the 36 outcomes sit**

|  | Value | Note |
| --- | --- | --- |
| Totals 6, 7 and 8 | 16 of 36 | 44% of all rolls land in these three |
| Totals 2 and 12 | 2 of 36 | Only 5.6% of rolls, one way each |
| Any double | 6 of 36 | 1+1 up to 6+6: one roll in 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.*

> **Adding the dice is not the same as rolling a single 2–12 die:** A single die that gave 2 to 12 evenly would make every total equally likely. Two dice do not, because several different pairs lead to the same total. The more dice you add, the more bell-shaped the totals become. See [the bell curve](https://toolcloak.pages.dev/articles/the-bell-curve-68-95-99-7-rule-z-scores-and-what-rare-really-means) for why.

## 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.

**Chance of at least one six in four rolls**

1. **Chance of no six on one roll.** 5 out of 6, which is about 0.833.
2. **Chance of no six in all four rolls.** (5/6)⁴ = 0.482, because the rolls are independent.
3. **Flip it around.** 1 − 0.482 = 0.518, so a six turns up 51.8% of the time.

**Out of 100 games of each bet, how many are won?**

| Group | Count |
| --- | --- |
| Win: a six in 4 rolls | 51.8% |
| Lose | 48.2% |

*A small edge for the player, enough to win steadily over many games.*

**The double-six bet in 24 rolls**

| Group | Count |
| --- | --- |
| Win: a double six in 24 rolls | 49.1% |
| Lose | 50.9% |

*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%.

**Chance of a run of heads in a row, from the start (%)**

|  | Chance |
| --- | --- |
| 1 head | 50% |
| 2 heads | 25% |
| 3 heads | 12.5% |
| 5 heads | 3.13% |
| 8 heads | 0.39% |
| 10 heads | 0.1% |

*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.*

> **The gambler’s fallacy:** Believing that a long streak must be corrected by the opposite outcome is called the gambler’s fallacy. A famous casino example from 1913 had the ball land on black 26 times in a row. Many players kept betting red, because it was “due”. Each spin was still close to 50:50 (a little less, because of the green zero).

## 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:

**Share of heads after more and more flips (%)**

| Number of flips | Share of heads | Fair value 50% |
| --- | --- | --- |
| 10 | 40 | 50 |
| 50 | 58 | 50 |
| 100 | 57 | 50 |
| 500 | 51.4 | 50 |
| 1,000 | 50.6 | 50 |
| 5,000 | 49.8 | 50 |

*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

1. What is the chance of a total of 9 on two dice? 4 of 36, which is **11.1%**.
2. Which two totals are equally likely as 5? **9**, since both have 4 ways.
3. After four tails in a row, what is the chance the next flip is heads? **50%**.
4. 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](https://toolcloak.pages.dev/articles/can-two-uuids-ever-be-the-same-the-odds-in-pictures) and [password entropy](https://toolcloak.pages.dev/articles/password-entropy-bits-guesses-and-why-length-beats-symbols).

## Frequently asked questions

### 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.