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Statistics & Data Science · pattern A, Numeric fields

Binomial probabilities, exactly and cumulatively.

Enter the number of trials, successes and the probability per trial for the exact probability, the cumulative probability, and the mean and variance of the distribution.

Inputs

The formula used

P(X = k) = ⁿCₖ p^k (1 − p)^(n−k)

Requires a fixed number of independent trials, each with the same probability of success and only two outcomes. If trials are not independent, the binomial does not apply.

P(X = k)

0.120134

Combinations ⁿCₖ
125,970
Mean, np
10.000
Variance, np(1−p)
5.0000
Standard deviation
2.2361
P(X ≤ k)
0.868412
P(X ≥ k)
0.251722

Exact and cumulative answer different questions.

Exam questions usually ask for at least k or at most k rather than exactly k, and the cumulative sum is where marks are lost — particularly the at least case, which needs one minus the sum up to k − 1, not up to k. Write out which tail you are summing before computing anything. When n is large and p is small, the Poisson distribution approximates the binomial and is much easier by hand.

Questions about binomial distribution

When can I use a normal approximation?
When np and n(1 − p) both exceed about five. Apply a continuity correction of ±0.5 or the approximation is noticeably off for small n.
What if trials are not independent?
Then the binomial is the wrong model — sampling without replacement from a small population needs the hypergeometric distribution instead.
Why is ⁿCₖ in the formula?
Because there are ⁿCₖ different orderings that produce exactly k successes, and each has the same probability. The coefficient counts the arrangements.

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