1 · Observe

Each bar is P(X = k): the chance of exactly k successes in n trials.

2 · Manipulate

3 · Predict

Set n = 10, p = 0.5. Before looking: which k is most likely, and roughly how likely is it? Now drag p to 0.9 — does the peak move? Does the spread change?

4 · See

5 · Explain

Each bar comes from P(X = k) = C(n,k)·p^k·(1−p)^n−k: choose which k trials succeed, then multiply the chances. The center sits at the mean μ = np and the spread is σ = √(np(1−p)) — widest at p = 0.5, squeezed near p = 0 or 1.