The binomial distributionEdexcel International A Level Further Maths: Revision notes

A binomial distribution $\mathrm{B}(n,p)$ models the number of successes $X$ in $n$ trials provided:

• there is a fixed number of trials, $n$;
• each trial has just two outcomes, success or failure;
• the probability of success $p$ is the same for every trial;
• the trials are independent. Write $X\sim\mathrm{B}(n,p)$ and $q=1-p$. Example: the number of sixes in $10$ rolls of a fair die is $\mathrm{B}\left(10,\frac16\right)$.

$\mathrm{P}(X=r)=\binom nr p^r(1-p)^{n-r},\quad r=0,1,\dots,n.$ The coefficient $\binom nr=\frac{n!}{r!(n-r)!}$ counts the arrangements of the $r$ successes among the $n$ trials. Example: $X\sim\mathrm{B}(8,0.4)$. $\mathrm{P}(X=3)=\binom83(0.4)^3(0.6)^5=56\times0.064\times0.07776=0.279$.

Tables give $\mathrm{P}(X\le r)$. Convert every probability into this form:

• $\mathrm{P}(X\ge r)=1-\mathrm{P}(X\le r-1)$
• $\mathrm{P}(X>r)=1-\mathrm{P}(X\le r)$
• $\mathrm{P}(a\le X\le b)=\mathrm{P}(X\le b)-\mathrm{P}(X\le a-1)$ Example: $X\sim\mathrm{B}(8,0.4)$. $\mathrm{P}(X\ge5)=1-\mathrm{P}(X\le4)=1-0.8263=0.1737$ and $\mathrm{P}(3\le X\le5)=0.9502-0.3154=0.6348$. Tables stop at $p=0.5$. For $p>0.5$, let $Y=n-X\sim\mathrm{B}(n,1-p)$. Example: $X\sim\mathrm{B}(10,0.7)$, $\mathrm{P}(X\le4)=\mathrm{P}(Y\ge6)=1-0.9527=0.0473$. A calculator can give the same values.

For $X\sim\mathrm{B}(n,p)$, no derivation needed: $\mathrm{E}(X)=np,\qquad\mathrm{Var}(X)=np(1-p).$ The standard deviation is $\sqrt{np(1-p)}$. The variance is always less than the mean. Example: $\mathrm{B}(30,0.4)$ has mean $12$ and variance $7.2$. To find $n$ and $p$ from the mean and variance, divide: $\frac{np(1-p)}{np}=1-p$. If the mean is $12$ and the variance is $7.2$, then $1-p=0.6$, $p=0.4$ and $n=30$.

Say whether each condition is reasonable in the context.

• Independence may fail when trials affect each other, for example sampling without replacement from a small population, or members of a group influenced by one another.
• Constant $p$ may fail when the probability changes with time, person or batch.
• Two outcomes: a trial with several outcomes needs to be simplified to success or failure.
• Fixed $n$: the number of trials must not depend on the outcomes. Compare data with the model: for binomial counts the variance is smaller than the mean. If observed variance is much larger, trials are probably not independent or $p$ varies.