Probability simulator
Design and run repeated chance experiments and watch the maths happen. Pick coin flips, die rolls, a spinner, or drawing coloured counters from a bag, choose how many trials to run (from 10 up to 10,000), and see the relative frequency of each outcome climb toward the theoretical probability. A conditional-probability mode runs two dependent trials and compares the estimate with the exact calculated value. Everything is computed in code, with a seeded random generator so a demo can be repeated exactly. Free, no sign-up.
| Outcome | Count | Relative frequency | Theoretical |
|---|---|---|---|
| Heads | 0 | 0.0% | 50.0% |
| Tails | 0 | 0.0% | 50.0% |
Run more trials and watch each observed bar settle toward its dashed theoretical marker. This is the point of the standard: with only a few trials the results jump around, but over hundreds or thousands of trials the relative frequency closes in on the true probability.
What this shows: the law of large numbers
Every chance experiment has a theoretical probability you can work out on paper: a fair coin is 1/2 heads, a fair six-sided die is 1/6 for each face, a spinner with 5 equal sectors is 1/5 each. When you actually run the experiment, the relative frequency (how often an outcome really came up, divided by the number of trials) will not match the theory exactly, and with only a handful of trials it can be well off.
The pattern to look for is convergence: as the number of trials grows into the hundreds and thousands, the relative frequency settles down and closes in on the theoretical probability. That is the law of large numbers, and it is the whole idea behind using simulations to estimate probabilities you might not be able to calculate easily. Run 10 trials a few times and watch the bars jump around; run 10,000 and watch them line up with the dashed theoretical markers.
Conditional probability and dependent events
Two events are independent when the first does not change the second: rolling a 6 tells you nothing about the next roll, so the chance of a 6 next is still 1/6. The simulator will show the conditional estimate landing on the same value as the plain probability, which is how you confirm independence.
Drawing counters without replacement is the classic dependent case. Put 2 red and 1 blue counter in the bag. If your first counter is red, only 1 red and 1 blue are left, so the chance the second is red is now 1/2, not the original 2/3. Set that up in the tool, switch to conditional mode, and compare the simulated estimate with the exact calculated value.
How to use it in class
- Start with a coin at 10 trials and run it several times, so students see how much small samples vary.
- Jump to 1,000 and then 10,000 trials and point out how close each bar gets to its dashed marker.
- Switch to a die or a spinner to show the same idea with more than two outcomes.
- Use the bag with 2 red and 1 blue, without replacement, then open conditional mode to model P(second red given first red) = 1/2.
- Tick Fixed seed and share the seed so every student's screen shows the same run, ideal for a whole-class discussion.
Common questions
- What is relative frequency?
- Relative frequency is how often an outcome actually happened divided by the number of trials, e.g. 47 heads out of 100 flips is a relative frequency of 0.47. As you run more trials it settles toward the theoretical probability (0.5 for a fair coin). This simulator lets you watch that happen live.
- Why don't 10 flips give exactly 5 heads?
- With only a few trials, chance variation is large, so 10 flips might give 3 heads or 7. That is normal. The law of large numbers says the relative frequency gets closer to the true probability as the number of trials grows, which is exactly why the tool lets you jump from 10 up to 10,000 trials.
- What is conditional probability?
- Conditional probability is the chance of one event given that another has already happened, written P(B given A). Drawing counters without replacement is the classic example: once you have taken a red counter out, the bag has changed, so the chance the next counter is red is different. The tool estimates this by simulation and compares it with the exact calculated value.
- Are the results really random?
- Each run reseeds a pseudo-random number generator with fresh entropy, so results differ every time and feel random. If you want a repeatable classroom demo, tick 'Fixed seed' and everyone running the same seed and settings gets the same results.
Curriculum links
This tool is the digital-tools fulfilment for the Australian Curriculum (v9) probability standards that call for repeated chance experiments and simulations:
- AC9M5P02 – simulations and variation (Year 5)
- AC9M6P02 – simulations and expected results (Year 6)
- AC9M7P02 – experiments and simulations (Year 7)
- AC9M8P03 – simulate compound events (Year 8)
- AC9M10P02 – model conditional probability (Year 10)
See more free classroom tools.