Repeat the same chance.

A binomial variable counts successes in a fixed number n of independent trials, each with the same success probability p and only two categories. “Success” simply means the category being counted.

To get exactly r successes, multiply the probability of a particular success/failure sequence by the number of such sequences, nCr. Use a sum for a range, or a complement for “at least one”. The mean is np and the variance is np(1 − p).

INTERACTIVE MODEL

Try it. Watch it change.

Number of trials n8 Success probability p0.4

P(X = 1) = 0.0896 · P(X ≤ 1) = 0.1064
Mean = 3.2 · variance = 1.92

Select a bar to read its exact model probability. The probabilities sum to one; bars represent discrete outcomes.

Explore: Change n and p. Select a bar to compare a single probability with the cumulative probability.

Binomial probability

P(X=r)=\binom nr p^r(1-p)^{n-r}

WORKED EXAMPLE

X ∼ Bin(4, 0.5). Find $P(X=2)$ .

$P(X=2)$ = 4C2(0.5)²(0.5)².
This is $\frac{6}{16}$ = 0.375.

Assumed knowledge

Combinations, independent trials and probability distributions.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Recognise the assumptions of a binomial model.
Calculate probabilities and distribution summaries.

QCAA Mathematical Methods 2025 v1.3 · p. 26 ↗

READY TO TRY IT?

Repeat the same chance.

X ∼ Bin(4, 0.5). Find P(X=2)P(X=2)P(X=2).

Put the idea to work.

X ∼ Bin(4, 0.5). Find $P(X=2)$ .