What is the binomial distribution, and how do you use the formula?

A footballer scores 80% of her penalties. If she takes 5 penalties, how likely is she to score exactly 4? Each penalty is a trial with two outcomes, success (goal) or failure (miss). The number of goals, X, can be 0, 1, 2, 3, 4 or 5, and the binomial distribution gives the probability of each value.

You can use it when four conditions hold. A memory aid is BINS:

• Binary: each trial has only two outcomes, success or failure.
• Independent: one trial does not affect another.
• Number of trials is fixed: you know n in advance.
• Same probability: the chance of success p is the same in every trial.

Then we write X ~ B(n, p): "X follows a binomial distribution with n trials and success probability p". The formula is

P(X = k) = ⁿCₖ × pᵏ × (1 − p)ⁿ⁻ᵏ

Each part has a job. pᵏ is the chance of k successes, (1 − p)ⁿ⁻ᵏ is the chance of the n − k failures, and ⁿCₖ ("n choose k") counts how many different orders those successes can come in. For 5 penalties with exactly 4 goals, the miss could be the 1st, 2nd, 3rd, 4th or 5th penalty, so ⁵C₄ = 5. You can find ⁿCₖ with the nCr button on a calculator, from Pascal's triangle, or with n! ÷ (k!(n − k)!).

So P(X = 4) = 5 × 0.8⁴ × 0.2¹ = 5 × 0.4096 × 0.2 = 0.4096.

Three more facts are used all the time:

• Mean (expected value): μ = np. Out of 5 penalties she expects 5 × 0.8 = 4 goals on average.
• Variance: σ² = np(1 − p), and the standard deviation is σ = √(np(1 − p)). See standard deviation.
• Cumulative probability: P(X ≤ k) adds up P(X = 0) to P(X = k). Tables and calculators give these directly.

The binomial distribution is discrete: X can only be a whole number from 0 to n. It is symmetric when p = 0.5, skewed to the right when p is small and skewed to the left when p is close to 1. When n is large and p is not too close to 0 or 1, its shape gets close to the bell curve of the normal distribution.

What is the binomial distribution used for?#

For counting successes in repeated yes-or-no trials: goals from penalties, faulty items in a batch, correct guesses in a test, patients who respond to a treatment. It is also the basis of many simple hypothesis tests.

Is the binomial distribution discrete or continuous?#

Discrete. X counts successes, so it can only be 0, 1, 2, … up to n. The normal distribution, by contrast, is continuous.

How do you use a binomial distribution table?#

Most tables give cumulative probabilities P(X ≤ k) for chosen n and p. Find your n and p, then read the row for k. For P(X = k), subtract: P(X ≤ k) − P(X ≤ k − 1). For P(X ≥ k), use 1 − P(X ≤ k − 1).

How do you do the binomial distribution on a TI-84?#

Press 2nd, VARS (DISTR). binompdf(n, p, k) gives P(X = k), and binomcdf(n, p, k) gives P(X ≤ k). For X ~ B(5, 0.8), binompdf(5, 0.8, 4) returns 0.4096.

How do you find the binomial distribution in Excel?#

=BINOM.DIST(k, n, p, FALSE) gives P(X = k), and =BINOM.DIST(k, n, p, TRUE) gives P(X ≤ k). For example, =BINOM.DIST(4, 5, 0.8, FALSE) gives 0.4096.

What is the negative binomial distribution?#

It turns the question around: instead of fixing the number of trials and counting successes, it counts how many trials (or failures) you need before reaching a fixed number of successes. It is used, for example, for "how many calls until the third sale?".