Probability Simulator

About probability simulation

A probability simulator runs a random experiment, such as flipping a coin or rolling dice, thousands of times and tallies how often each outcome appears. It is a hands-on way to watch the law of large numbers at work: over a short run the observed frequencies wander all over the place, but as the trial count grows they settle toward the fixed theoretical probability. A fair coin should land heads 50 percent of the time, a single six-sided die should show each face one-sixth of the time, and the sum of two dice follows a triangular distribution that peaks at 7.

The phrase "snake eyes" comes from craps and refers to rolling two ones on a pair of dice, the lowest possible total. There is exactly one way to roll it out of 36 equally likely combinations, so its probability is 1/36, roughly 2.78 percent. The simulator lets you confirm that figure empirically: roll the 2-dice mode a few thousand times and watch the count for the total of 2 converge toward that 1-in-36 rate.

How it works

Each trial uses the browser's pseudo-random generator to pick an outcome with equal weighting, then increments a counter. The observed probability of any outcome is simply its count divided by the total number of trials.

Theoretical probability  P(outcome) = favourable outcomes / total outcomes
Observed probability      f(outcome) = count(outcome) / N trials
Law of large numbers      f(outcome) -> P(outcome)  as N -> infinity
Snake eyes (two dice = 2)  P = 1/36 = 2.78%
Any single die face        P = 1/6  = 16.67%
Fair coin (heads)          P = 1/2  = 50%

The expected absolute error shrinks in proportion to 1 over the square root of N. That is why ratios after 10 trials can be wildly off, yet after 1,000 or more they look stable to within a percentage point or two.

Worked example

Suppose you select the 2-dice mode and click +1000 four times, for 4,000 total rolls. You are watching the total of 7, the most likely sum.

1. Favourable outcomes for a 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) which is 6 combinations.
2. Theoretical probability: 6 / 36 = 0.1667, or 16.67 percent.
3. Expected count in 4,000 rolls: 4,000 x 0.1667 = about 667.
4. Typical observed range: roughly 630 to 700, since random noise of a few percent is normal at this sample size.

Two-dice probability reference

Rolling two six-sided dice gives 36 equally likely combinations. The table shows how many ways each total can occur and its probability.

Common pitfalls

• Expecting balance too soon. After 10 coin flips a 7-to-3 split is completely normal. Convergence needs hundreds or thousands of trials, not dozens.
• The gambler's fallacy. A run of five heads does not make tails "due". Each flip is independent, so the next flip is still 50/50 regardless of history.
• Confusing the count gap with the percentage gap. As trials grow the raw difference between heads and tails counts can widen even while the percentage gap shrinks toward zero.
• Assuming all dice totals are equally likely. A 7 is six times more likely than a 2 with two dice, because more combinations add up to 7.
• Reading a pseudo-random generator as truly random. Browser randomness is good enough for games and demos but is not cryptographically secure.
• Stopping at a lucky moment. If you halt the run exactly when a frequency happens to match theory, you are cherry-picking rather than observing genuine convergence.

Related tools

Frequently asked questions