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

Calculating binomial probabilities

Throughout this section we will use the random variable $X~\mathrm{B}(n,p)$. For binomial, the probability of 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.

How do I find a probability from a binomial distribution, P(X = x)?

• You should have a calculator that can calculate binomial probabilities

• You want to use the "Binomial Probability Distribution" function

  • This is sometimes shortened to BPD, Binomial PD or Binomial Pdf

• You will need to enter:

  • The '$x$' value - the value of x for which you want to find $\mathbf{P}\mathbf{(}\mathit{X}\mathbf{=}\mathit{x}\mathbf{)}$

  • The '$n$' value - the number of trials

  • The '$p$' value - the probability of success

• Some calculators will give you the option of listing the probabilities for multiple values of $x$ at once

• There is a formula that you can use but you are expected to be able to use the distribution function on your calculator

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

    • If there are $x$successes then there are $(n-x)$ failures

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

    • This can be seen by considering a probability tree diagram with n trials, where p is the probability of success and the tree diagram is being used to find x successes

    •  $\left(\begin{array}{c}n\\ x\end{array}\right)$ is the number of pathways through the tree there would be exactly x successes within the n trials

  • You might find it quicker to use the formula than finding using the binomial probability distribution function on your calculator

How do I find a cumulative probability from a binomial distribution, P(X ≤ x)?

• You should have a calculator that can calculate cumulative binomial probabilities

  • Most calculators will only find $\mathit{P}(\mathit{X}\le \mathit{x})$

  • Some calculators can find $\mathrm{P}(a\le X\le b)$

• You want to use the "Binomial Cumulative Distribution" function

  • This is sometimes shortened to BCD, Binomial CD or Binomial Cdf

• You will need to enter:

  • The 'x' value - the value of x for which you want to find $\mathit{P}\mathbf{(}\mathit{X}\mathbf{\le }\mathit{x}\mathbf{)}$

    • Some will instead ask for lower and upper bounds

    • For this lower would be 0 and upper would be x

  • The '$n$' value - the number of trials

  • The '$p$' value - the probability of success

How do I find P(X ≥ x)?

• You might be lucky enough to have a calculator that has lower and upper bounds:

  • Use $\mathit{x}$ for the lower bound and $\mathit{n}$ for the upper bound

  • Otherwise, you will need some extra identities

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

  • This is 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)?

• You might be lucky enough to have a calculator that has lower and upper bounds:

  • Use a for the lower bound and b for the upper bound

  • Otherwise, you will need some extra identities

• $a\le X\le b$: 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 probability of X  is zero for non-integer values 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: $P(4\le X\le 9)=P(X\le 9)-P(X\le 3)$

How do I find P(X<x) and P(X>x)?

• 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)

How do I use the binomial cumulative distribution function tables?

• In your formula booklet you get tables which list the values of P(X ≤ x)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 0.05, 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

  • If you want more accurate values then you will need to use your calculator

• The tables are useful when you want to find a value of x given the probability

  • For example, the largest value of such that P(X ≤ x) is less than 0.95

• You can estimate P(X = k  )using the tables by using:

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

  • To get a more accurate estimate use the formula or the binomial probability distribution function on your calculator

• 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:

    • $P(X=k)=P(Y=n-k)$

    • $P(X\le k)=P(Y\ge n-k)$

    • $P(X\ge k)=P(Y\le n-k)$