A-level · Beta

Binomial distribution

Move the success probability, select outcomes and build cumulative events.

Drag P on the probability track. Select a bar or use the outcome slider to choose k.

P(X=k)=\binom nk p^k(1-p)^{n-k}

Tab to a ring or outcome, then use the arrow keys. Sliders also work with touch and keyboard.

Shaded bars make up the selected event. The dashed green line marks np.

Use this model for a fixed number of independent trials, two outcomes and a constant success probability. Decimal probabilities are approximate.

Predict, test and explain

Step 1 of 3

Predict

If X is binomial with n = 4 and p = ½, what is P(X = 2)?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

If X is binomial with n = 4 and p = ½, what is P(X = 2)?

$X\sim\mathrm{Bin}(4,\tfrac12),\quad P(X=2)$

Hint

There are a fixed number of independent trials with constant p.

Reveal answer

There are 6 ways to choose two successes; each specific outcome has probability 1/16.

$\frac38$

Hint 1

There are a fixed number of independent trials with constant p.

Hint 2

For exactly k successes, use n choose k times pᵏ(1 − p)ⁿ⁻ᵏ.

Hint 3

A cumulative event adds several disjoint outcome probabilities.

Worked example

This example uses fixed values, separate from the diagram controls.

Find P(X = 2) for X ~ Bin(4, ½).

Answer: 3/8.

Connect this idea

Watch out: The mean np = 2 is a value, not a probability.

Specification and learning route

AQA N1 · Edexcel Statistics 4.1

AS and A-level. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Probability; Independent trials; Combinations; Discrete random variables.