Binomial Distribution

This lesson covers the binomial distribution, one of the most important discrete probability distributions at A-Level. The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

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Conditions for a Binomial Distribution

A random variable $X$X follows a binomial distribution, written $X \sim B(n, p)$X∼B(n,p), if:

1. There is a fixed number of trials, $n$n.
2. Each trial has exactly two outcomes: success or failure.
3. The probability of success, $p$p, is constant for each trial.
4. The trials are independent of each other.

Exam Tip: When asked to justify a binomial model, you must state all four conditions and relate them to the context of the problem. Simply listing the conditions without context will not gain full marks.

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The Binomial Probability Formula

The probability of exactly $r$r successes in $n$n trials is:

$P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}$P(X=r)=(rn​)pr(1−p)n−r

where $\binom{n}{r} = \frac{n!}{r!(n-r)!}$(rn​)=r!(n−r)!n!​ is the binomial coefficient (number of ways to choose $r$r items from $n$n).

Example: A fair coin is tossed 8 times. Find $P(X = 3)$P(X=3) where $X$X is the number of heads.