Calculating Binomial Probabilities (Edexcel IAL Maths)

Spec point — 1.1

Calculating Binomial Probabilities

Throughout this section we will use the random variable $X \sim \mathrm{B} ( n , p )$. 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 $Y \sim B ( n , 1 - p )$ if the probability of success is bigger than 0.5

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

    • $P(X=k)=P(Y=n-k)$
    • $P(X\leq k)=P(Y\geq n-k)$
    • $P(X\geq k)=P(Y\leq n-k)$

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

  • $P ( X = x ) = ( \begin{gathered} n \\ x \end{gathered} ) p^{x ( 1 - p )^{n} - x}$

    • The number of times this can happen is calculated by the binomial coefficient$( \begin{gathered} n \\ x \end{gathered} ) = C pre^{n}_{x} = \frac{n factorial}{x \text{factorial} ( n - x ) factorial}$

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

  • $P(X=k)=P(X\leq k)-P(X\leq k-1)$

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 $Y \sim B ( n , 1 - p )$

  • $P(X\leq x)=P(Y\geq n-x)$

How do I find P(X ≥ x)?

• $X\geq x$: 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

• $P(X\geq x)=1-P(X<x)$

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

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

  • $P(X\geq x)=1-P(X\leq x-1)$
  • For example: $P(X\geq10)=1-P(X\leq9)$

How do I find P(a ≤ X ≤ b)?

• $a\leq X\leq b$: 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

• $P(a\leq X\leq b)=P(X\leq b)-P(X<a)$

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

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

  • $P(a\leq X\leq b)=P(X\leq b)-P(X\leq a-1)$
  • For example: $P(4\leq X\leq9)=P(X\leq9)-P(X\leq3)$

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

  • $P(X<x)=P(X\leq x-1)$ and $P(X>x)=P(X\geq x+1)$
  • For example: $P(X<5)=P(X\leq4)$ and $P(X>5)=P(X\geq6)$
  • 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, : $P(4<X\leq10)$

  • You want the integers 5 to 10
  • You want the integers up to 10 excluding the integers up to 4
  • $P(X\leq10)-P(X\leq4)$

• 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)

Worked Example The random variable $X \sim \mathrm{B} ( 40 , 0.35 )$ . Find:

(a) $P(X=10)$

(b) $P(X\leq10)$

(c) $P(X\geq10)$

(d) $P(8<X<10)$

Answer:

Exam Hint

• Some calculators will calculate probabilities for binomial distributions
• These are great for checking your answers once you have answered your question showing the appropriate method