Free dice probability simulator. Roll 2d6 thousands of times and watch the sum distribution converge to the bell curve - see why 7 is most common.
This dice simulator rolls two six-sided dice (2d6) anywhere from ten to one hundred thousand times and plots the results as they come in. Instead of a single roll, you watch the distribution of sums build up — and you see the messy early data smooth out into a clean, predictable shape. It is the difference between knowing the odds and watching them emerge.
With two dice, the sum can be anything from 2 to 12, but those totals are not equally likely. There is only one way to roll a 2 (1+1) and one way to roll a 12 (6+6), but there are six different ways to roll a 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). So a 7 is six times more likely than a 2. Plotted across all eleven possible sums, the probabilities form a symmetric triangular distribution peaking at 7 — exactly the dashed "expected" overlay the simulator draws on top of your live results.
| Sum | Ways | Probability |
|---|---|---|
| 2 or 12 | 1 | 2.78% |
| 3 or 11 | 2 | 5.56% |
| 4 or 10 | 3 | 8.33% |
| 5 or 9 | 4 | 11.11% |
| 6 or 8 | 5 | 13.89% |
| 7 | 6 | 16.67% |
A single die is uniform — every face is equally likely, so its distribution is flat. Yet the moment you add two dice together, the flat shape becomes a triangle that bulges in the middle. Add more dice and that triangle rounds further into the familiar bell curve. This is the central limit theorem: sums of independent random variables tend toward a normal distribution, no matter how the individual variables are shaped. The simulator lets students and teachers see that abstract theorem turn into a concrete, animated graph.
Run a small sample first — say 20 rolls — and the bars look jagged and lopsided. Run 10,000 and they snap almost perfectly onto the expected triangle. That convergence is the law of large numbers made visible. Export the chart as a PNG or copy the embed code to drop the live simulation into a lesson, a blog post, or a study guide.
Because there are more combinations that sum to 7 than to any other total. Six different pairs (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) produce a 7, while only one pair produces a 2 or a 12. That gives 7 a 16.67% probability, the highest of any sum.
The sum of 2d6 follows a symmetric triangular distribution over the values 2 through 12, peaking at 7. It is not uniform: the middle sums are far more likely than the extremes, because more combinations produce them.
A single die is uniform (flat). Adding dice together makes the sum cluster in the middle: 2d6 forms a triangle, and more dice round it toward a bell curve. This shows the central limit theorem, which says sums of independent random variables approach a normal distribution.
A few dozen rolls look jagged and uneven. By 1,000 to 10,000 rolls the observed bars settle closely onto the theoretical triangle. This convergence of observed frequency toward true probability is the law of large numbers in action.
Yes. You can run the simulation live in class, adjust the number of rolls and the animation speed, export the chart as a PNG image, or copy an embed snippet to place the interactive simulator directly in a lesson, slide, or blog post.