CIE A Level Maths: Probability & Statistics 1

Revision Notes

3.2.2 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 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.

Where does the formula for a binomial distribution come from?

  • The formula for calculating an individual binomial probability is
    • begin mathsize 16px style straight P left parenthesis X equals r right parenthesis equals p subscript r equals open parentheses table row n row r end table close parentheses p to the power of r open parentheses 1 minus p close parentheses to the power of n minus r end exponent end style
      • If there are r successes then there are open parentheses n minus r close parentheses failures
      • The number of times this can happen is calculated by the binomial coefficient
      • open parentheses table row n row r end table close parentheses equals C presuperscript n subscript r equals fraction numerator n factorial over denominator r factorial left parenthesis n minus r 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 r successes
  •  stretchy left parenthesis table row n row r end table stretchy right parenthesis is the number of pathways through the tree there would be exactly r successes within the n trials
  • The formula allows statisticians to quickly find probabilities for larger values of n without needing to draw the whole tree diagram
  • Your calculator may have a function that would allow you to calculate binomial probabilities
    • You can learn how to use this to check your work but it is important you always show your working using the formula to get the marks in the exam

How do I calculate the cumulative probabilities for a binomial distribution?

  • Most of the time you will be required to calculate cumulative binomial probabilities rather than individual ones
  • Use the formula to find the individual probabilities and then add them up
  • Make sure you are confident working with inequalities for discrete values
    • Only integer values will be included so it is easiest to look at which integer values you should include within your calculation
    • Sometimes it is quicker to find the probabilities that are not being asked for and subtract from one
  • begin mathsize 16px style straight P left parenthesis X less or equal than r right parenthesis end styleis asking you to find the probabilities of all values up to and including r
    • This means all values that are at most r
    • Don’t forget to include P(X = 0)
    • It could also be written as 
  • is asking you to find the probabilities of all values up to but not including r
    • This means all values that are less than r
    • Stop at r - 1
    • It could also be written as 
  • is asking you to find the probabilities of all values greater than and including r
    • This means all values that are at least r
    • It could also be written as 
  • is asking you to find the probabilities of all values greater than but not including r
    • This means all values that are more than r
    • Start at r + 1
    • It could also be written as 
  • If calculating begin mathsize 16px style straight P left parenthesis a less or equal than X less or equal than b right parenthesis end style pay attention to whether the probability of a and b should be included in the calculation or not
    • For example, :
      • You want the integers 5 to 10

Worked example

If X is the random variable X tilde straight B left parenthesis 10 comma 0.35 right parenthesis. Find:

(i)
straight P left parenthesis X equals 3 right parenthesis
(ii)
straight P left parenthesis X less or equal than 3 right parenthesis
(iii)
straight P left parenthesis X greater than 3 right parenthesis
(iv)
straight P left parenthesis 3 less than X less than 6 right parenthesis 

3-2-2-calculating-binomial-probabilities-we-solution

Exam Tip

  • Looking carefully at the inequality within the probability is key here, make sure you consider which integers should be counted within your calculations.

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