The binomial distributionEdexcel A-Level Maths: Revision notes

A binomial distribution $X\sim B(n,p)$ models the number of successes in $n$ trials when: there is a fixed number of trials $n$; each trial has two outcomes (success or failure); the probability of success $p$ is constant; the trials are independent. Examples: the number of correct guesses in a 10-question test with four options ($p=0.25$), or defective bolts in a sample of 20 ($p=0.05$). State the distribution with its parameters, and justify the model by naming these conditions.

For $X\sim B(n,p)$, $P(X=r)=\binom{n}{r}p^r(1-p)^{n-r},\qquad r=0,1,\dots,n.$ $\binom nr$ counts the ways of choosing which $r$ trials are successes. Example: $X\sim B(10,0.25)$ gives $P(X=3)=\binom{10}{3}(0.25)^3(0.75)^7=0.250$. Use the calculator's binomial probability density function for $P(X=r)$, and check you have not left out the $\binom nr$ factor.

The calculator's cumulative function gives $P(X\le r)$. Convert other inequalities to this form, using the fact that $X$ takes whole numbers only:

• $P(X<r)=P(X\le r-1)$
• $P(X\ge r)=1-P(X\le r-1)$
• $P(X>r)=1-P(X\le r)$
• $P(a\le X\le b)=P(X\le b)-P(X\le a-1)$. Example: $X\sim B(10,0.25)$, $P(X\ge4)=1-P(X\le3)=1-0.7759=0.224$. 'At least one' is quickest as $1-P(X=0)=1-(1-p)^n$.

Sometimes a binomial probability becomes the $p$ of a second binomial. Example: a sample is accepted if $X\le2$ with $q=P(X\le2)=0.9245$. The number accepted from 5 samples is $W\sim B(5,0.9245)$, so $P(W=4)=5q^4(1-q)=0.276$. To find the least $n$ for a target, form an inequality: $1-0.4^n>0.99$ gives $0.4^n<0.01$, so $n\log0.4<\log0.01$ and, dividing by the negative $\log0.4$, $n>5.03$, so $n=6$. Also find the largest $k$ with $P(X\ge k)>0.9$ by evaluating cumulative probabilities for neighbouring values.

Binomial answers are only reliable if the conditions hold. Independence may fail when trials share a cause, such as students in one class or consecutive throws affected by fatigue. The probability may vary between trials. Comment in context: 'independence is unlikely because ...' and say what difference it would make.