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The Binomial Distribution

Count the successes in n independent trials. On a Galton board every peg is a trial and every bin is a count — and the pile of balls takes the shape of C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ.

Step 1 of 12

Speed

A Galton board with 8 rows of pegs. A ball dropped at the top hits one peg per row and bounces left or right — each with probability 1/2, independently of every other bounce.

P(X = 4) = \binom{8}{4}\, 0.5^{4}\, 0.5^{4} = 70 \times 0.0625 \times 0.0625 = \mathbf{0.2734} \\[6pt] \text{observed: } \tfrac{0}{0} = 0, \qquad E[X] = np = 4, \; \operatorname{Var}(X) = np(1-p) = 2

n (rows of pegs)8p (chance of right)0.5Balls0

The Binomial Distribution

Count the successes in n independent trials. On a Galton board every peg is a trial and every bin is a count — and the pile of balls takes the shape of C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ.

Step 1 of 12

Speed

A Galton board with 8 rows of pegs. A ball dropped at the top hits one peg per row and bounces left or right — each with probability 1/2, independently of every other bounce.

P(X = 4) = \binom{8}{4}\, 0.5^{4}\, 0.5^{4} = 70 \times 0.0625 \times 0.0625 = \mathbf{0.2734} \\[6pt] \text{observed: } \tfrac{0}{0} = 0, \qquad E[X] = np = 4, \; \operatorname{Var}(X) = np(1-p) = 2

n (rows of pegs)8p (chance of right)0.5Balls0

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The Binomial Distribution

See also

The Binomial Distribution

See also