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 reaching for the calculator.
Builds on Conditional probability and The binomial expansion.
IN THIS TOPIC
- 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 between statements like “more than”, “at least” and calculator-friendly cumulative forms.
WHAT YOU PROBABLY THINK
Ten percent of a batch are faulty, so in a sample of 10 exactly one will be faulty.
When the 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 independent of one another. Then X, the number of successes, has the distribution X ~ B(n, p).
The conditions are the exam question. Sampling without replacement breaks constant p; one student's answer influencing another's breaks independence. When a condition fails only slightly, a large population sampled lightly, the binomial is still a reasonable model, and saying so in context is the mark.
Calculating with it
The formula is counting dressed as algebra: p to the x for the successes, (1 − p) to the rest for the failures, and the binomial coefficient for the number of orders it could happen in.
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 the answer should be the biggest single probability, and it is.
Cumulative probabilities come from the calculator's binomial CD function, and the translation table is worth rehearsing: 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 errors here cost more marks than the probability theory ever does.
THE EXAM BIT
- Checking the model means naming the conditions in context, not reciting them: say what a trial is, what success is, and why p stays constant.
- Write the distribution X ~ B(n, p) before calculating; the statement itself carries a mark.
- Translate inequalities into ≤ form before touching the calculator, and write the translation down.
- Answers to probability questions are decimals to three significant figures unless the exact fraction is natural.
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.
CHECK YOUR PROGRESS
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- 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 between statements like “more than”, “at least” and calculator-friendly cumulative forms.
No animated video for this topic yet; these notes stand alone.