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

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Calculating Binomial Probabilities

Throughout this section we will use the random variable X tilde straight B left parenthesis n comma p right parenthesis. 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 tilde B left parenthesis n comma 1 minus p right parenthesis if the probability of success is bigger than 0.5
    • Remember X plus Y equals n, which leads to identities:
      • begin mathsize 16px style straight P left parenthesis X equals k right parenthesis equals straight P left parenthesis Y equals n minus k right parenthesis end style
      • straight P left parenthesis X less or equal than k right parenthesis equals straight P left parenthesis Y greater or equal than n minus k right parenthesis
      • straight P left parenthesis X greater or equal than k right parenthesis equals straight P left parenthesis Y less or equal than n minus k right parenthesis

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 left parenthesis X equals x right parenthesis equals open parentheses table row n row x end table close parentheses space p to the power of x open parentheses 1 minus p close parentheses to the power of n minus x end exponent 
      • The number of times this can happen is calculated by the binomial coefficient open parentheses table row n row x end table close parentheses equals C presuperscript n subscript x equals fraction numerator n factorial over denominator x factorial left parenthesis n minus x right parenthesis factorial end fraction
  • You can also use the tables for the Binomial Cumulative Distribution Function in the formulae booklet
    • bold P bold left parenthesis bold italic X bold equals bold italic k bold right parenthesis bold equals bold P bold left parenthesis bold italic X bold less or equal than bold italic k bold right parenthesis bold minus bold P bold left parenthesis bold italic X bold less or equal than bold italic k bold minus bold 1 bold right parenthesis 

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 tilde B left parenthesis n comma 1 minus p right parenthesis
    • P left parenthesis X less or equal than x right parenthesis equals P left parenthesis Y greater or equal than n minus x right parenthesis

How do I find P(X ≥ x)?

  • X greater or equal than 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 left parenthesis X greater or equal than x right parenthesis equals 1 minus P left parenthesis X less than x right parenthesis
  • As x  is an integer then P left parenthesis X less than x right parenthesis equals P left parenthesis X less or equal than x minus 1 right parenthesis as the probability of X is zero for non-integer values for a binomial distribution
  • Therefore, to calculate P left parenthesis X greater or equal than x right parenthesis:
    • bold P bold left parenthesis bold italic X bold greater or equal than bold italic x bold right parenthesis bold equals bold 1 bold minus bold P bold left parenthesis bold italic X bold less or equal than bold italic x bold minus bold 1 bold right parenthesis
    • For example: P left parenthesis X greater or equal than 10 right parenthesis equals 1 minus P left parenthesis X less or equal than 9 right parenthesis

How do I find  P(a ≤ X ≤ b)?

  • a less or equal than X less or equal than 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 left parenthesis a less or equal than X less or equal than b right parenthesis equals P left parenthesis X less or equal than b right parenthesis minus P left parenthesis X less than a right parenthesis
  • As X is an integer then P left parenthesis X less than a right parenthesis equals P left parenthesis X less or equal than a minus 1 right parenthesis as the P left parenthesis X equals x right parenthesis equals 0  for non-integer value of x for a binomial distribution
  • Therefore to calculate P left parenthesis a less or equal than X less or equal than b right parenthesis:
    • bold italic P bold left parenthesis bold italic a bold less or equal than bold italic X bold less or equal than bold italic b bold right parenthesis bold equals bold italic P bold left parenthesis bold italic X bold less or equal than bold italic b bold right parenthesis bold minus bold italic P bold left parenthesis bold italic X bold less or equal than bold italic a bold minus bold 1 bold right parenthesis 
    • For example: P left parenthesis 4 less or equal than X less or equal than 9 right parenthesis equals P left parenthesis X less or equal than 9 right parenthesis minus P left parenthesis X less or equal than 3 right parenthesis

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
    • bold P bold left parenthesis bold italic X bold less than bold italic x bold right parenthesis bold equals bold P bold left parenthesis bold italic X bold less or equal than bold italic x bold minus bold 1 bold right parenthesis and bold P bold left parenthesis bold X bold greater than bold x bold right parenthesis bold equals bold P bold left parenthesis bold X bold greater or equal than bold x bold plus bold 1 bold right parenthesis
    • For example: straight P left parenthesis X less than 5 right parenthesis equals straight P left parenthesis X less or equal than 4 right parenthesis and P left parenthesis X greater than 5 right parenthesis equals P left parenthesis X greater or equal than 6 right parenthesis
    • 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, straight P left parenthesis 4 less than X less or equal than 10 right parenthesis
    • You want the integers 5 to 10
    • You want the integers up to 10 excluding the integers up to 4
    • P left parenthesis X less or equal than 10 right parenthesis minus P left parenthesis X less or equal than 4 right parenthesis
  • 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 tilde straight B left parenthesis 40 comma 0.35 right parenthesis . Find:

(a)
straight P left parenthesis X equals 10 right parenthesis
(b)
straight P left parenthesis X less or equal than 10 right parenthesis
(c)
straight P left parenthesis X greater or equal than 10 right parenthesis
(d)
straight P left parenthesis 8 less than X less than 10 right parenthesis
(a)
straight P left parenthesis X equals 10 right parenthesis
1-1-2-calculating-binomial-prob-we-solution-part-1
(b)
straight P left parenthesis X less or equal than 10 right parenthesis
1-1-2-calculating-binomial-prob-we-solution-part-2
(c)
straight P left parenthesis X greater or equal than 10 right parenthesis
1-1-2-calculating-binomial-prob-we-solution-part-3
(d)
straight P left parenthesis 8 less than X less than 10 right parenthesis
1-1-2-calculating-binomial-prob-we-solution-part-4

Examiner Tip

  • 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

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.