Quick guide
- Use the simulator to build intuition and compare theory vs runs.
- Open the streak page when the next question is consecutive wins or losses.
- Open expected value when the next question is average payoff, loss, or return per trial.
- Use the tree / 2×2 table to structure AND/OR/NOT.
- Use nCr/nPr for combinations and permutations.
Tools
- Distribution Sampler and Histogram Tool.
Generate random samples from common distributions, visualize histograms, review summary stats, and export CSV/JSON.
- Randomness Tests (chi-square, runs, ACF, JB).
Run chi-square, runs, autocorrelation, and Jarque-Bera tests on bit or number sequences.
- Shuffle bias comparison – naive vs Fisher–Yates.
Compare shuffle bias: naive swap-everywhere vs Fisher–Yates (and optional sort(() => random - 0.5)).
- Random Walk & Markov Chain Visualizer.
Run random walk (1D/2D/3D) and Markov chain simulations in your browser.
- Dice stats (odds & distribution) + PNG sharing.
Compute odds and distributions for NdS +/- K.
- Dice roller (d4 to d100, 2d6+3, PNG).
Roll d4 to d100 or use notation like 2d6+3.
- TRPG dice check (d20/d100, pools) with PNG sharing.
Run d20/d100 checks with advantage/disadvantage, compare vs DC, and roll dice pools to count successes.
- Random number generator (integers, decimals, seed).
Generate random numbers in your browser: integers or decimals, optional seed, unique mode, and one-click copy.
Calculators
- Probability Simulator — coin, dice, roulette.
Simulate coin tosses, dice rolls and roulette spins with a fixed seed to compare theoretical and empirical.
- Streak Probability Simulator.
Estimate the chance of consecutive wins or losses, expected streak starts, and likely longest streaks in one.
- Expected Value Simulator.
Compare theoretical expected value, variance, and simulated averages for coin flips, dice, or simple lottery-style.
- Probability tree & 2×2 table — AND/OR & complement.
Probability tree & 2×2 table tool for AND/OR/NOT and complements.
- Birthday paradox (collision probability) calculator.
Calculate the birthday paradox collision probability with exact math + a Monte Carlo simulator.
- Coupon Collector Calculator (Expected Draws & Completion).
Solve the coupon collector problem with expected draws n·H_n, completion probability P(T≤t), and t50/t90/t99.
- Distributions — binomial, Poisson, t, chi square.
Compute PMF or PDF, CDF, quantiles, and exact intervals for binomial, Poisson, Student t, and chi-square.
- Hypergeometric probability calculator without replacement.
Calculate hypergeometric probabilities without replacement, including P(X=10) for N=60, K=35, n=15.
- nCr & nPr calculator (permutations & combinations).
Compute nCr (combinations), nPr (permutations), and factorial (n!).
How to choose the right probability tool first
Start from the question, not the formula. Use this page to decide whether you need simulation, exact counting, a distribution model, or a one-off probability story such as birthdays or coupon collection.
Open the simulator first when you want intuition
Use Probability Simulator or the sampling tools when you want to see how outcomes spread over repeated runs. This is the best first stop for coin tosses, dice, roulette, random walks, and "what happens if I repeat this many times?" questions.
Use formulas first when the structure is fixed
Open nCr / nPr for counting arrangements, Probability tree / 2×2 for AND/OR/NOT logic, and Distributions when you already know the model family. This route is faster when you need an exact probability rather than a simulated estimate.
When to use a special-purpose calculator
- Use Coupon collector when the question is "how many draws until I probably finish the set?"
- Use Birthday paradox when the question is about collision or duplicate risk in a group.
- Use Hypergeometric when draws are without replacement.
Common mistakes on this topic
- Using simulation when the exact count is easy and faster to explain.
- Using a binomial-style model when the process is actually without replacement.
- Comparing weighted random choices against equal-chance results without stating the weighting rule.
- Reading one simulated run as a guarantee instead of a rough distribution of outcomes.
See also
FAQ
Which probability tool should I open first?
Open the simulator when you want intuition from repeated runs, a tree or 2×2 table when you need AND/OR/NOT logic, and nCr/nPr when the question is about counting arrangements. If the process already names a model such as binomial, Poisson, or t, go straight to the distributions page.
When should I use a simulator instead of a formula?
Use a simulator when you want to see spread, rare outcomes, or repeated-run intuition. Use a formula first when the setup is fixed and you need an exact probability or exact count that is easier to report.
How do I know whether weighted and equal chance are different problems?
If each item has the same chance on every draw, use an equal-chance tool. If entries, probabilities, or transition weights differ by item, use a weighted picker, a weighted sampler, or a Markov/random-walk tool that keeps those weights explicit.
Why are coupon collector and birthday paradox separate calculators?
They answer different repeated-chance questions. Coupon collector is about how long it takes to collect all categories, while birthday paradox is about the chance of at least one collision in a group.
Are these pages good for study as well as quick answers?
Yes. The topic is arranged so you can start with a quick answer tool, then move to trees, distributions, or simulators to understand why the result changes under different assumptions.