Calculating Binomial Probabilities (Edexcel International A Level (IAL) Maths: Statistics 2): Revision Note

Calculating Binomial Probabilities

Throughout this section we will use the random variable $X~\mathrm{B}(n,p)$. For binomial, the probability of a X  taking a non-integer or negative value is always zero. Therefore any values mentioned in this section will be assumed to be non-negative integers.

What are the tables for the binomial cumulative distribution function?

• In your formulae booklet you get tables which list the values of for different values of x, p and n

  • $n$ can be 5, 6, 7, 8, 9 10, 12, 15, 20, 25, 30, 40, 50

  • $p$ can be 05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5

  • $x$ can be different values depending on n

• The probabilities are rounded to 4 decimal places

• The values of $p$ only go up to 0.5

  • You can instead count the number of failures $Y~B(n,1-p)$ if the probability of success is bigger than 0.5

  • Remember $X+Y=n$, which leads to identities:

    • $\mathrm{P}(X=k)=\mathrm{P}(Y=n-k)$

    • $\mathrm{P}(X\le k)=\mathrm{P}(Y\ge n-k)$

    • $\mathrm{P}(X\ge k)=\mathrm{P}(Y\le n-k)$

How do I calculate, P(X = x) the probability of a single value for a binomial distribution?

• You can use the formula given in the formulae booklet
  

  • $P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{\left(1-p\right)}^{n-x}$ 
    

    • The number of times this can happen is calculated by the binomial coefficient $\left(\begin{array}{c}n\\ x\end{array}\right)={}^{n}C_x=\frac{n!}{x!(n-x)!}$

• You can also use the tables for the Binomial Cumulative Distribution Function in the formulae booklet
  

  • $\mathbf{P}\mathbf{(}\mathit{X}\mathbf{=}\mathit{k}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{(}\mathit{X}\mathbf{\le }\mathit{k}\mathbf{)}\mathbf{-}\mathbf{P}\mathbf{(}\mathit{X}\mathbf{\le }\mathit{k}\mathbf{-}\mathbf{1}\mathbf{)}$ 

How do I calculate, P(X ≤ x), the cumulative probabilities for a binomial distribution?

• If x is small, you could find the probability of each possible value of x and then add them together

• Otherwise, you will have to use the tables for the Binomial Cumulative Distribution Function in the formulae booklet

• If p is bigger than 0.5 then you will have to use the number of failures $Y~B(n,1-p)$

  • $P(X\le x)=P(Y\ge n-x)$

How do I find P(X ≥ x)?

• $X\ge x$: This means all values of X which are at least x

  • These are all values of X except the ones that are less than x

• $P(X\ge x)=1-P(X<x)$

• As x  is an integer then $P(X<x)=P(X\le x-1)$ as the probability of X is zero for non-integer values for a binomial distribution

• Therefore, to calculate $P(X\ge x)$:

  • $\mathbf{P}\mathbf{(}\mathit{X}\mathbf{\ge }\mathit{x}\mathbf{)}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{P}\mathbf{(}\mathit{X}\mathbf{\le }\mathit{x}\mathbf{-}\mathbf{1}\mathbf{)}$

  • For example: $P(X\ge 10)=1-P(X\le 9)$

How do I find  P(a ≤ X ≤ b)?

• $a\le X\le b$: This means all values of X which are at least a and at most b

  • This is all the values of X which are no greater than b except the ones which are less than a

• $P(a\le X\le b)=P(X\le b)-P(X<a)$

• As X is an integer then $P(X<a)=P(X\le a-1)$ as the $P(X=x)=0$  for non-integer value of x for a binomial distribution

• Therefore to calculate $P(a\le X\le b)$:

  • $\mathit{P}\mathbf{(}\mathit{a}\mathbf{\le }\mathit{X}\mathbf{\le }\mathit{b}\mathbf{)}\mathbf{=}\mathit{P}\mathbf{(}\mathit{X}\mathbf{\le }\mathit{b}\mathbf{)}\mathbf{-}\mathit{P}\mathbf{(}\mathit{X}\mathbf{\le }\mathit{a}\mathbf{-}\mathbf{1}\mathbf{)}$ 

  • For example: $P(4\le X\le 9)=P(X\le 9)-P(X\le 3)$

What if an inequality does not have the equals sign (strict inequality)?

• For a binomial distribution (as it is discrete) you could rewrite all strict inequalities (< and >) as weak inequalities (≤ and ≥) by using the identities for a binomial distribution

  • $\mathbf{P}\mathbf{(}\mathit{X}\mathbf{<}\mathit{x}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{(}\mathit{X}\mathbf{\le }\mathit{x}\mathbf{-}\mathbf{1}\mathbf{)}$ and $\mathbf{P}\mathbf{(}\mathbf{X}\mathbf{>}\mathbf{x}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{(}\mathbf{X}\mathbf{\ge }\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{)}$

  • For example: $\mathrm{P}(X<5)=\mathrm{P}(X\le 4)$ and $P(X>5)=P(X\ge 6)$

  • Though it helps to understand how they work

• It helps to think about the range of integers you want

• Always find the biggest integer that you want to include and the biggest integer that you then want to exclude

• For example, : $\mathrm{P}(4<X\le 10)$

  • You want the integers 5 to 10

  • You want the integers up to 10 excluding the integers up to 4

  • $P(X\le 10)-P(X\le 4)$

• For example, P(X > 6)  :

  • You want the all the integers from 7 onwards

  • You want to include all integers excluding the integers up to 6

  • 1- P(X ≤ 6)

• For example, P(X < 8)  :

  • You want the integers 0 to 7

  • P(X ≤ 7)