Trillion - Probability Simulator
Trillion is a probability sandbox. You describe one or more events, such as a 1 in 6 chance, a 0.3% drop rate or a 25% success rate, choose how many trials to run, and the simulator rolls the dice for you and counts how often each event happened.
Numbers like 'one in a thousand' are hard to feel. Watching what actually comes out of ten trials, a thousand trials and ten million trials makes the law of large numbers concrete: small samples bounce around, large samples settle close to the true probability. It is handy for students, teachers, game designers checking drop rates and anyone curious about chance.
The simulator runs entirely in your browser with no account. Your events and past runs are kept in this browser.
The app runs entirely in your browser. What you enter is saved only on this device.
How to use Trillion
- Name an event. In Add New Probability Event, type a short name such as 'Six' or 'Rare drop'.
- Enter its probability. Choose Decimal (0 to 1), Percentage or Fraction. Type 0.25 or 25, or for a fraction enter the numerator and denominator separately, such as 1 and 4. Tap Add Event. Repeat for more events; each gets its own colour.
- Set the number of trials. In Simulation Settings enter any whole number from 1 up to 1,000,000,000,000.
- Run the simulation. Tap Run Simulation. A progress message appears, then the Simulation Results card lists each event with its count and observed percentage.
- Compare with the expected value. Multiply the probability by the number of trials to get the expected count and see how close the run came. Run again to see how much the result varies.
- Review history. Open the Simulation History tab to see earlier runs with their trial counts and results.
Features
- Several probability events in one simulation
- Probability input as decimal, percentage or fraction
- Up to one trillion trials per run
- Trial-by-trial simulation up to 100 million trials
- Normal approximation for runs above 100 million trials
- Observed counts and percentages for every event
- History of past simulations
- Events and history saved in your browser
Understanding simulated probability
Expected value and variation
If an event has probability p and you run n independent trials, the expected number of successes is n × p. The actual count will almost never match exactly. Its typical spread is described by the standard deviation, √(n × p × (1 − p)). For a fair coin tossed 100 times, the expected count is 50 and the standard deviation is 5, so results between about 40 and 60 are completely normal.
As n grows, the spread grows more slowly than n itself, so the observed percentage gets closer to the true one. That is the law of large numbers in action.
Rare events need lots of trials
A 1 in 1,000 chance does not mean you are guaranteed a success in 1,000 tries. The probability of at least one success in n tries is 1 − (1 − p)^n. For p = 1/1000 and n = 1000 that is only about 63%. Even after 3,000 tries there is roughly a 5% chance of seeing none. Simulating a rare event a few times makes this very clear and is a good antidote to the gambler's fallacy, the belief that a result is 'due'.
How several events are handled
In each simulated trial a single random number decides which of your events happened, so the events behave like mutually exclusive outcomes, such as the faces of a die or the rarities in a loot table. If your probabilities add up to more than 1, they are scaled down proportionally so they fit. If they add up to less than 1, the remainder is simply 'nothing happened'.
Exact simulation versus approximation
Up to 100 million trials, every trial is actually simulated with a pseudo-random number. Beyond that, simulating each trial would take far too long, so Trillion uses the normal approximation to the binomial distribution: it draws a count centred on n × p with the appropriate spread. For very large n this is statistically very close to doing every trial, but results for extremely rare events (where n × p is small) are less precise.
Frequently asked questions
How long does a run take?
Small runs are instant. Runs of tens of millions of trials take a few seconds in a browser because each trial is simulated, and the page may not respond until it finishes. Very large runs above 100 million use an approximation and finish quickly.
Are the results truly random?
They use the browser's pseudo-random number generator, which is fine for exploring probability but not suitable for cryptography or anything with real stakes.
Where is my data stored?
Your events and simulation history are saved only in this browser on this device. There is no account. Clearing site data or using another browser starts fresh.
Can I simulate dependent events?
No. Each trial is independent and the events within a trial are treated as mutually exclusive outcomes.
What is different from the phone app?
The simulator works the same. The iPhone and Android apps show ads that a subscription removes; the web version has no ads or purchases.
Limitations of the web version
- Large exact runs can make the page unresponsive for a few seconds
- Runs above 100 million trials use a statistical approximation
- No charts; results are shown as counts and percentages
- Data stays in one browser and does not sync
Phone app
Trillion is also available as an app for phones and tablets: