A small choice, repeated
At every row, a virtual ball chooses right with probability p and left with probability 1 − p. Choices are independent with the same p. The final bin counts how many right turns occurred.
How can random choices make a predictable shape?
Release a few virtual balls, then a thousand. Change the chance of a right turn and compare the results with a mathematical prediction.
Enable JavaScript to change the conditions and run the interactive experiment.
Make a discovery
One random outcome can surprise you. Many independent outcomes reveal a distribution. A prediction describes the chances of different results; it does not promise an exact count in one run.
Make a prediction
A fair independent coin has just landed heads five times. What is its chance of heads next?
- Less than half; tails is due
- Still one half
- More than half; the streak continues
Read the explanation
Independence means previous tosses do not change the next probability. A long-run pattern is not a balancing rule for the very next event.
Understand it
The shape has a name
The number of right turns follows a binomial distribution. More paths lead to middle counts than to the extremes when p is one half. Biasing each choice shifts the distribution; it need not remain symmetric.
More trials reveal the pattern
The colored bars are observed counts. Gold ticks are expected counts for the selected sample size. A larger sample often gives closer proportions, but another seed can move them away again. Chance does not owe you a correction after a streak.
Look closer at the science
Count all the paths
For n independent choices, P(X = k) = C(n,k)pᵏ(1 − p)ⁿ⁻ᵏ. C(n,k) counts the possible orders of k right turns. Probabilities sum to one. At n = 4 and p = 0.5, bin probabilities are 1, 4, 6, 4, 1 divided by 16.
Center and spread
E[X] = np and Var(X) = np(1 − p). The expected number in bin k is sample size × P(X = k). Expected need not mean most likely, and an expected count can be fractional.
Randomness you can replay
A versioned 32-bit pseudorandom generator supplies uniform draws, compared with p. A fixed seed and configuration reproduce the same choices. This is a mathematical Bernoulli board, not a physical collision simulation or cryptographic randomness.
Try it yourself: Make a tiny chance experiment
Supplies
- 10 equal paper slips
- A cup or bowl
- A pencil and tally sheet
- Build a known proportion
Mark five slips R and five L. Fold them alike, put all ten in the cup and mix. Predict how many R results you will get in twenty draws.
- Draw, replace, mix
Without looking, draw one slip, tally the letter, return it to the cup and mix again. Repeat twenty times. Replacement keeps the composition fixed; mixing helps avoid a predictable draw.
- Compare several runs
Count R outcomes. Repeat another twenty draws if you like. Compare each run and the combined proportion with one half. Discuss why an imperfect mix or distinguishable slips can change the experiment.
Will a 50% chance give exactly half successes in twenty tries?
Paper draws approximate independent random sampling. Keep small slips away from very young children; an adult can handle drawing. The result does not prove fairness.
Sources and model limits
- Each row is an independent Bernoulli choice. A real pegboard can have correlated impacts, bounce dynamics and uneven geometry; those are not modeled.
- The seed is deterministic, and changing it chooses another repeatable run. This does not establish that a physical coin is fair.
- No gambling advice, forecasts or inferential confidence intervals. Long-run convergence does not mean each next trial compensates for past outcomes.
Independent events retain the same probability conditional on previous results.
Introductory Statistics 2e §3.2, independence and sampling examples. Supports the streak question and replacement/mixing rationale.
OpenStax · IndependenceIndependent fixed-probability binary trials produce a binomial count.
Introductory Statistics 2e §4.3, binomial conditions, formula, mean and standard deviation. Supports bin probabilities and moments.
OpenStax · Binomial distributionExperimental relative frequencies vary, and probabilities describe long-run behavior.
Introductory Statistics 2e §3.1, probability terminology and law of large numbers. Supports the distinction between predicted chance and observed count.
OpenStax · Probability basicsIndependent subject review is pending.
Read the sources and model assumptions
