The binomial distributionAQA A-Level Maths: Revision notes

A random variable $X$ has a binomial distribution, written $X\sim B(n,p)$, when it counts the number of successes in $n$ trials and these conditions hold:

• there is a fixed number $n$ of trials;
• each trial has two outcomes, success or failure;
• the probability of success $p$ is the same for every trial;
• the trials are independent. Example: 10 seeds, each germinating with probability $0.3$ independently, gives $X\sim B(10,0.3)$. State the conditions in the context, such as 'each seed germinates with the same probability, independently of the others'.

For $X\sim B(n,p)$, the probability of exactly $r$ successes is $P(X=r)=\binom{n}{r}p^r(1-p)^{n-r},$ where $\binom nr$ counts the arrangements of $r$ successes among $n$ trials. For $X\sim B(10,0.3)$: $P(X=3)=\binom{10}{3}(0.3)^3(0.7)^7=120\times0.027\times0.0823543=0.267$. The probability distribution is the set of all $P(X=r)$, which add to 1. The calculation of the mean and variance of $X$ is not needed at this level; concentrate on probabilities.

Use the binomial probability function for $P(X=r)$ and the cumulative binomial function for $P(X\le r)$. Enter $n$, $p$ and $r$. Cumulative functions only give 'at most', so rewrite other probabilities in that form:

• $P(X<r)=P(X\le r-1)$
• $P(X\ge r)=1-P(X\le r-1)$
• $P(X>r)=1-P(X\le r)$
• $P(a\le X\le b)=P(X\le b)-P(X\le a-1)$ Example: $X\sim B(20,0.25)$. $P(X>7)=1-P(X\le7)=1-0.898=0.102$. For $X\sim B(12,0.15)$: $P(2\le X\le4)=P(X\le4)-P(X\le1)=0.9761-0.4435=0.533$.

The probability from a binomial calculation can itself be used in another probability. If a box is rejected with probability $p=0.135$ and 5 boxes are independent, $P(\text{at least one rejected})=1-(1-p)^5=0.516$. If 12 patients are booked and there are 10 slots, more patients attend than slots when at most 1 does not attend: $P(X\le1)=0.443$. Watch the wording: decide what counts as a success before choosing $p$. 'Does not attend' might be success with $p=0.15$, so 'more than 10 attend' becomes $X\le1$.

The binomial distribution is a model: real situations only approximately meet its conditions. Evaluate the model by questioning the conditions. In a clinic, patients from one family may attend or miss together, so independence may not hold. Quality-control examples may fail the constant probability condition if a machine drifts over time. If the conditions are not met, the calculated probabilities may be inaccurate, so state the limitation and, where possible, the likely effect. A model using a sample can also estimate $p$ from real data, but that value then carries its own uncertainty.