Probability Explorer
Interactive binomial distribution visualiser.
Visualize the binomial distribution and watch how probability outcomes spread as you change the number of trials and the success probability. An interactive way to build intuition for distributions, variance, and the Central Limit Theorem.
The binomial distribution answers this question: "If I repeat a yes/no experiment N times, where each trial has probability P of success, how likely is it that I get exactly K successes?"
Examples: flipping a coin 100 times and asking how likely is exactly 53 heads. Shooting 20 free throws at 75% and asking how likely is missing 4. Testing 50 products at 2% defect rate.
Frequently Asked Questions
What is a binomial distribution? ▾
A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. For example: flipping a coin 100 times and counting heads. The distribution shows how likely each possible count is.
When does the normal approximation apply? ▾
The normal approximation to the binomial distribution becomes accurate when both np ≥ 5 and n(1-p) ≥ 5. This is a consequence of the Central Limit Theorem, as n grows, the binomial distribution approaches a normal (bell curve) shape regardless of p.
What does the simulation overlay show? ▾
Running a simulation generates actual random trials and overlays the results on the theoretical distribution. With small sample sizes, the empirical and theoretical bars differ noticeably. As you increase the number of simulated trials, they converge, demonstrating the Law of Large Numbers.
What is the difference between probability and statistics? ▾
Probability works forward: given a known process, what outcomes should we expect? Statistics works backward: given observed outcomes, what can we infer about the underlying process? The binomial distribution sits at the intersection, it is a probability model used constantly in statistical inference.