Maths › Statistics › The binomial distribution
The binomial distribution
Count the successes in a fixed number of independent yes-or-no trials and the answer has a distribution you can write down. The binomial is the exam's favourite model, and the skill is checking it applies before you reach for the calculator.
Builds on Conditional probability and The binomial expansion.
IN THIS TOPIC
- Use a discrete probability distribution given as a table or as a formula, including finding an unknown constant.
- Recognise and name the discrete uniform distribution, and say why a given situation is or is not one.
- State and check the four conditions for X ~ B(n, p) to model a situation.
- Calculate P(X = x) from the formula and cumulative probabilities from a calculator.
- Convert “more than”, “at least” and “fewer than” into the ≤ form a calculator will accept.
COMMON MISCONCEPTION
Ten percent of a batch are faulty, so in a sample of 10 exactly one will be faulty.
Discrete distributions in general
A discrete random variable takes one of a listed set of values, each with a probability, and a probability distribution is that list. It can arrive as a table, or as a rule such as P(X = x) = kx² for x = 1, 2, 3. Whichever form it takes, the probabilities total 1, and that single fact answers most of the questions asked about it.
For P(X = x) = kx² on x = 1, 2, 3, summing gives k(1 + 4 + 9) = 1, so k = 1/14. Then P(X ≥ 2) = (4 + 9)/14 = 13/14. Note what is not asked. This specification excludes the mean and variance of a general discrete random variable, so a question wanting E(X) is a Further Maths question, not this one.
One named distribution belongs here. When every outcome is equally likely the variable has a discrete uniform distribution: on n listed values each carries probability 1/n, which is the fair die, the fair coin and the raffle ticket drawn from a hat. The specification asks you to recognise and name it, not to compute with it, so the work is spotting equal probabilities and saying so.
The word doing the work is equally. A die scoring 1 to 6 is uniform; the total of two dice is not, because 7 has six ways of happening and 2 has one. Where a question gives you a distribution and asks what model it is, uniform and binomial are the two names expected, and the check is whether the probabilities are all the same.
When the binomial model applies
A situation earns the binomial label when four things hold. A fixed number of trials n. Each trial with exactly two outcomes. A constant probability of success p. And trials that are independent of one another. Then X, the number of successes, has the distribution X ~ B(n, p).
The conditions are the exam question far more often than the arithmetic is. Sampling without replacement breaks constant p. One student copying another breaks independence. When a condition fails only slightly, as with a large population sampled lightly, the binomial stays a reasonable model, and saying that in context is where the mark sits.
Calculating with it
The formula is counting in algebraic clothing. p to the x for the successes, (1 − p) to the rest for the failures, and the binomial coefficient for the number of orders in which it could have happened.
WORKED EXAMPLE
A point probability by hand
A biased coin lands heads with probability 0.3. It is flipped 10 times. Find the probability of exactly 3 heads.
P(X = 3) = ¹⁰C₃ × 0.3³ × 0.7⁷ = 120 × 0.027 × 0.0823543.
That multiplies to 0.267 (3 s.f.).
Sense check: np = 3 is the typical count, so this should be the largest single probability in the distribution, and it is.
Cumulative probabilities come from the calculator's binomial CD function, and the translation is worth rehearsing until it is automatic. P(X < 5) is P(X ≤ 4). P(X ≥ 7) is 1 − P(X ≤ 6). P(3 ≤ X < 8) is P(X ≤ 7) − P(X ≤ 2). Off-by-one slips here cost more marks every year than the probability theory does.
ASSESSMENT FOCUS
- Checking the model means naming the conditions in this context. Say what one trial is, what counts as a success, and why p stays constant.
- Write X ~ B(n, p) before you calculate. The statement itself carries a mark.
- Translate every inequality into ≤ form on paper before touching the calculator, and leave the translation visible.
- Give probabilities to three significant figures unless the working produces an exact fraction.
CHECK YOURSELF
X ~ B(20, 0.15). Write P(X ≥ 5) in a form a calculator's cumulative function accepts, and state the value of P(X = 0) as a power.
Show a hint
At least five means not four-or-fewer.
Show the answer
P(X ≥ 5) = 1 − P(X ≤ 4).
P(X = 0) = 0.85²⁰, every trial failing at once.
Fixed n, two outcomes, constant p, independent trials. Only then is it binomial.
Point probabilities by the formula, everything else through ≤ and the cumulative function.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the the binomial distribution questions page.
CHECK YOUR PROGRESS
Rate how confident you feel with each objective for this lesson. Ratings are saved in this browser, on this device, unless you sign in.
- Use a discrete probability distribution given as a table or as a formula, including finding an unknown constant.
- Recognise and name the discrete uniform distribution, and say why a given situation is or is not one.
- State and check the four conditions for X ~ B(n, p) to model a situation.
- Calculate P(X = x) from the formula and cumulative probabilities from a calculator.
- Convert “more than”, “at least” and “fewer than” into the ≤ form a calculator will accept.
Open the full revision checklist to see every objective in the course in one place.