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The binomial distribution
The binomial distribution models the number of successes in a fixed number of independent trials, each with two outcomes and a constant probability of success. Check those conditions against the context, then calculate point and cumulative binomial probabilities.
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.
The number faulty is a random variable, not a fixed count. Ten fixed trials, two outcomes each, a constant p = 0.1 and independence give X ~ B(10, 0.1), and P(X = 1) = 10 × 0.1 × 0.99 = 0.387. One fault is the likeliest single outcome, and more than half of such samples contain some other number.
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, which determines any unknown constant.
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. Pearson Edexcel 9MA0 does not require the mean or variance of a general discrete random variable; E(X) is Further Mathematics content.
When every outcome is equally likely the variable has a discrete uniform distribution: on n listed values each has probability 1/n, as for a fair die, a fair coin or a raffle ticket drawn from a hat. Pearson Edexcel 9MA0 requires this distribution to be recognised and named rather than used in calculation.
The defining condition is that the probabilities are equal. A die scoring 1 to 6 is uniform; the total of two dice is not, because 7 arises in six ways and 2 in one. To identify the model in a given distribution, compare the probabilities: equal probabilities indicate a discrete uniform distribution.
When the binomial model applies
The binomial model requires four conditions: a fixed number of trials n; exactly two outcomes per trial; a constant probability of success p; and independent trials. When these hold, X, the number of successes, has the distribution X ~ B(n, p).
Check the conditions against the context. Sampling without replacement makes p vary between trials. One student copying another breaks independence. Where a condition fails only slightly, as when a small sample is taken from a large population, the binomial remains a reasonable model; state this in context when using it.
Calculating with it
The three factors are: p to the power x for the successes, (1 − p) to the power n − x for the failures, and the binomial coefficient for the number of orders in which those successes can occur.
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.).
Check: np = 3, so P(X = 3) is expected to be the largest single probability in this distribution, which it is.
Cumulative probabilities come from the calculator's binomial cumulative function, which evaluates P(X ≤ x). Convert each inequality to that form first: P(X < 5) is P(X ≤ 4); P(X ≥ 7) is 1 − P(X ≤ 6); and P(3 ≤ X < 8) is P(X ≤ 7) − P(X ≤ 2). Since X is discrete, the strict and non-strict inequalities differ by one whole value.
ASSESSMENT FOCUS
- To check the model, state the conditions in this context: what one trial is, what counts as a success, and why p is constant.
- State X ~ B(n, p) before calculating.
- Write every inequality in ≤ form before using the calculator, and leave that conversion 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, the probability that every trial is a failure.
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 binomial distribution questions page.
CHECK YOUR PROGRESS
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- 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.