What are the conditions for a binomial distribution?

A fixed number of identical trials, two possible outcomes in each, a constant probability of success, and independence between trials.

Do I need the binomial distribution table?

No. The curriculum notes say using the binomial table is not required; probabilities are calculated from the formula.

Why is the variance of a binomial distribution less than its mean?

Because Var(X) = np(1 − p) is the mean np multiplied by (1 − p), which is less than 1. This is one of the clues for telling a binomial from a Poisson distribution.

The Binomial Distribution (DSE M1)

If an experiment has n independent trials, each a success with the same probability p, the number of successes X follows a binomial distribution, X ~ B(n, p), with P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ, E(X) = np and Var(X) = np(1 − p).

Solve The Binomial Distribution questions with AI

B(6, p) as p moves from 0.1 to 0.9. At p = 0.3 the tallest bar is at k = 2, with height 0.3241 as in Example 1. Try it yourself →

Binomial Distribution Formulas

ProbabilityP(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ, k = 0, 1, …, n

MeanE(X) = np

VarianceVar(X) = np(1 − p)

"At least one"P(X ≥ 1) = 1 − (1 − p)ⁿ

When Is a Distribution Binomial? The Four Conditions

A fixed number n of identical trials.
Each trial has only two outcomes: success or failure.
The probability of success p is the same in every trial.
The trials are independent.

Check all four before writing X ~ B(n, p). Drawing without replacement from a small group fails the third and fourth conditions.

Worked Examples — The Binomial Distribution

Example 1: Exactly k successes

A biased coin shows heads with probability 0.3. It is tossed 6 times. Find the probability of getting exactly 2 heads, and the mean and variance of the number of heads.

Let X be the number of heads: X ~ B(6, 0.3)
P(X = 2) = C(6, 2)(0.3)²(0.7)⁴ = 15 × 0.09 × 0.2401
= 0.3241
E(X) = 6 × 0.3 = 1.8, Var(X) = 6 × 0.3 × 0.7 = 1.26

Answer: P(X = 2) = 0.3241, E(X) = 1.8, Var(X) = 1.26

Example 2: At least one success

For the same coin, find the probability of getting at least one head in 6 tosses.

P(X ≥ 1) = 1 − P(X = 0)
= 1 − (0.7)⁶ = 1 − 0.117649
= 0.8824

Answer: 0.8824

💡 Exam Traps & Secrets

⚠️ "At least", "at most", "more than""At least 2" is X ≥ 2; "more than 2" is X ≥ 3; "at most 2" is X ≤ 2. For "at least" it is usually quicker to use the complement: 1 − P(X = 0) − P(X = 1).

⚠️ State the distributionDefine the random variable and write X ~ B(n, p) before calculating. It shows the method and makes it obvious to you which n and p each part uses.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Binomial probability)X ~ B(8, 0.4). Find P(X = 3) and P(X ≤ 1), and show the distribution as a bar chart.

(At least)A multiple-choice test has 10 questions with 4 options each. Find the probability of guessing at least 3 correctly.

(Mean and variance)X ~ B(n, 0.2) and E(X) = 4. Find n and Var(X).