The Binomial Distribution

You’ll be able to

Verify the four conditions (BINS) for a binomial setting
Compute binomial probabilities and the mean and standard deviation
Distinguish "exactly," "at most," and "at least" probability requests

The binomial setting: BINS

A count X is binomial when four conditions hold — remember BINS: Binary (each trial is a success or failure), Independent trials, a Number of trials n that is fixed in advance, and the Same probability of success p on every trial. X counts the successes in the n trials. If any condition fails — especially a fixed n — the model does not apply.

Computing binomial probabilities

The probability of exactly k successes multiplies three things: the number of arrangements of k successes among n trials, the probability of the k successes, and the probability of the n − k failures. Watch the phrasing: "at most k" means P(X ≤ k) (add up 0 through k), "at least k" means P(X ≥ k) (often easiest as 1 minus the smaller tail), and "exactly k" is a single term.

Binomial probability, mean, and standard deviation

P(X = k) = C(n,k) · p^k · (1−p)^(n−k) · μ_X = np · σ_X = sqrt(np(1−p))

C(n,k) is the number of ways to choose which k of the n trials are successes. The mean is np; the standard deviation is the square root of np(1−p).

Worked example

A free-throw shooter makes 80% of shots. In 10 independent attempts, find the expected number made and the standard deviation, then set up P(exactly 8 made).

1.Confirm binomial: binary (make/miss), independent shots, fixed n = 10, same p = 0.8. BINS holds.
2.Mean: μ = np = 10(0.8) = 8 made.
3.Standard deviation: σ = sqrt(np(1−p)) = sqrt(10 · 0.8 · 0.2) = sqrt(1.6) ≈ 1.26.
4.P(X = 8) = C(10,8)·(0.8)^8·(0.2)^2 = 45·(0.8)^8·(0.2)^2 ≈ 0.302.

Answer: Expected makes μ = 8, standard deviation σ = sqrt(1.6) ≈ 1.26 shots, and P(exactly 8) = C(10,8)(0.8)^8(0.2)^2 ≈ 0.302.

Checkpoint

Which situation is a valid binomial setting?

Watch out

Read the boundary words carefully. "At least 8" is P(X ≥ 8) = P(8) + P(9) + P(10), while "at most 8" is P(X ≤ 8). Mixing these up is a frequent error; when in doubt, list exactly which values of X the phrase includes.

Checkpoint

For a binomial random variable with n = 50 and p = 0.2, what is the standard deviation?

Answer the 2 checkpoints as you read.