AQA A Level Maths: Statistics

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4.2.2 Calculating Binomial Probabilities

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Calculating Binomial Probabilities

Throughout this section we will use the random variable begin mathsize 16px style X tilde straight B left parenthesis n comma space p right parenthesis end style. 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.

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

  • 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 'begin mathsize 16px style x end style' value - the value of  for which you want to find bold P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis
    • 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 begin mathsize 16px style x end style of 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 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 
      • If there xare successes then there are left parenthesis n minus x right parenthesis failures
      • 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
      • 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
      •  open parentheses table row n row x end table close parentheses 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 calculate, P(X ≤ x) , the cumulative probabilities for a binomial distribution?

  • You should have a calculator that can calculate cumulative binomial probabilities
    • Most calculators will only find begin mathsize 16px style bold italic P left parenthesis bold italic X less or equal than bold italic x right parenthesis end style
    • Some calculators can find begin mathsize 16px style straight P left parenthesis a less or equal than X less or equal than b right parenthesis end style
  • 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 for which you want to find begin mathsize 16px style bold italic P bold left parenthesis bold italic X bold less or equal than bold italic x bold right parenthesis end style
      • Some will instead ask for lower and upper bounds
      • For this lower would be 0 and upper would be x
    • The 'begin mathsize 16px style n end style' value - the number of trials
    • The 'begin mathsize 16px style p end style' 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 begin mathsize 16px style bold italic x end style for the lower bound and begin mathsize 16px style bold italic n end style for the upper bound
    • Otherwise, you will need some extra identities
  • begin mathsize 16px style X greater or equal than x end style: 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
  • begin mathsize 16px style P left parenthesis X greater or equal than x right parenthesis equals 1 minus P left parenthesis X less than x right parenthesis end style
  • As x  is an integer then begin mathsize 16px style P left parenthesis X less than x right parenthesis equals P left parenthesis X less or equal than x minus 1 right parenthesis end style as the probability of X is zero for non-integer values for a binomial distribution
  • Therefore to calculate begin mathsize 16px style P left parenthesis X greater or equal than x right parenthesis end style:
    • begin mathsize 16px style 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 end style
    • For example: begin mathsize 16px style 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 end style

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
  • begin mathsize 16px style a less or equal than X less or equal than b end style: This means all values of 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
  • begin mathsize 16px style 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 end style
  • As x is an integer then size 16px P size 16px left parenthesis size 16px X size 16px less than size 16px a size 16px right parenthesis size 16px equals size 16px P size 16px left parenthesis size 16px X size 16px less or equal than size 16px a size 16px minus size 16px 1 size 16px right parenthesis as the probability of X is zero for non-integer values for a binomial distribution
  • Therefore to calculate begin mathsize 16px style P left parenthesis a less or equal than X less or equal than b right parenthesis end style:
    •  
    • For example: size 16px P size 16px left parenthesis size 16px 4 size 16px less or equal than size 16px X size 16px less or equal than size 16px 9 size 16px right parenthesis size 16px equals size 16px P size 16px left parenthesis size 16px X size 16px less or equal than size 16px 9 size 16px right parenthesis size 16px minus size 16px P size 16px left parenthesis size 16px X size 16px less or equal than size 16px 3 size 16px 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
    •  begin mathsize 16px style 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 end style and 
    • For example:  and size 16px P size 16px left parenthesis size 16px X size 16px greater than size 16px 5 size 16px right parenthesis size 16px equals size 16px P size 16px left parenthesis size 16px X size 16px greater or equal than size 16px 6 size 16px 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, begin mathsize 16px style straight P left parenthesis 4 less than X less or equal than 10 right parenthesis end style
    • You want the integers 5 to 10
    • You want the integers up to 10 excluding the integers up to 4
    • begin mathsize 16px style P left parenthesis X less or equal than 10 right parenthesis minus P left parenthesis X less or equal than 4 right parenthesis end style
  • 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 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 space less than space X space less than space 15 right parenthesis
(a)
straight P left parenthesis X equals 10 right parenthesis
4-2-2-clalculating-binomial-probabilities-we-solution-part-1
(b)
straight P left parenthesis X less or equal than 10 right parenthesis
4-2-2-clalculating-binomial-probabilities-we-solution-part-2
(c)
straight P left parenthesis X greater or equal than 10 right parenthesis
4-2-2-clalculating-binomial-probabilities-we-solution-part-3
(d)
straight P left parenthesis 8 space less than space X space less than space 15 right parenthesis
4-2-2-clalculating-binomial-probabilities-we-solution-part-4

Exam Tip

  • Always make sure you are using the correct function on your calculator. Most questions will be in context so try and pick out the key words and numbers. If the question is worth more than one mark then be sure to show a method to get at least one mark if you write the answer down incorrectly.

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Dan

Author: Dan

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.