Count the successes in a fixed number of independent yes/no trials and you get a binomial random variable. Only two numbers control it — n, how many trials, and p, the chance of success on each one. Drag them and watch the whole distribution reshape.
Binomial (n = 10, p = 0.50)
P(X = k)in the highlighted regionk itselfsimulatedμ = npnormal curve
Binomial (n = 10, p = 0.50)drag the sliders below
The two parameters
10
0.50
Mean μ = np5.00
SD σ = √(npq)1.58
Most likely value5
Shape skew0.00
Try a shape
Read a probability off the chart
5
A count is binomial only if all four hold: the number of trials n is fixed in advance · each trial is independent · each trial has exactly two outcomes · and p is the same on every trial.
Does it actually happen that way?
Each "experiment" runs 10 independent trials and records how many successes came up. The gold bars are what really happened; the blue bars are what the formula predicted.
Experiments0
Mean successes—theory 5.00
SD of successes—theory 1.58
Run a batch and the gold bars will wobble around the blue ones. Keep running and the wobble shrinks — that is the long-run frequency the probabilities were describing all along.