Calculating Binomial Probabilities (DP IB Maths: AI HL)

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Calculating Binomial Probabilities

Throughout this section we will use the random variable X tilde straight B left parenthesis n comma space p right parenthesis. For binomial, the probability of X taking a non-integer or negative value is always zero. Therefore any values of X 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 GDC 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 straight P left parenthesis X equals x 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 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 GDC
    • straight P left parenthesis X equals x right parenthesis equals straight C presuperscript n subscript x cross times p to the power of x left parenthesis 1 minus p right parenthesis to the power of n minus x end exponent
      • straight C presuperscript n subscript x equals fraction numerator n factorial over denominator r factorial left parenthesis n minus r right parenthesis factorial end fraction

How do I calculate P(a X b): the cumulative probabilities for a binomial distribution? 

  • You should have a GDC that can calculate cumulative binomial probabilities
    • Most calculators will find straight P left parenthesis a less or equal than X less or equal than b right parenthesis
    • Some calculators can only find straight P left parenthesis X less or equal than b right parenthesis
      • The identities below will help in this case
  • You should use the "Binomial Cumulative Distribution" function
    • This is sometimes shortened to BCD, Binomial CD or Binomial Cdf
  • You will need to enter:
    • The lower value - this is the value a
      • This can be zero in the case straight P left parenthesis X less or equal than b right parenthesis
    • The upper value - this is the value b
      • This can be n in the case straight P left parenthesis X greater or equal than a right parenthesis
    • The 'n' value - the number of trials
    • The 'p' value - the probability of success

How do I find probabilities if my GDC only calculates P(≤ x)?

  • To calculate P(Xx) just enter x into the cumulative distribution function
  • To calculate P(X < x) use:
    • straight P left parenthesis X less than x right parenthesis equals straight P left parenthesis X less or equal than x minus 1 right parenthesis which works when is a binomial random variable
      • P(X < 5) = P(≤ 4)

  • To calculate P(X > x) use:
    • straight P left parenthesis X greater than x right parenthesis equals 1 minus straight P left parenthesis X less or equal than x right parenthesis which works for any random variable
      • P(X > 5) = 1 - P(≤ 5)
  • To calculate P(Xx) use:
    • straight P left parenthesis X greater or equal than x right parenthesis equals 1 minus straight P left parenthesis X less or equal than x minus 1 right parenthesis which works when is a binomial random variable
      • P(X ≥ 5) = 1 - P(≤ 4)
  • To calculate P(a Xb) use:
    • straight P left parenthesis a less or equal than X less or equal than b right parenthesis equals straight P left parenthesis X less or equal than b right parenthesis minus straight P left parenthesis X less or equal than a minus 1 right parenthesis which works when is a binomial random variable
      • P(5 ≤ ≤ 9) = P(≤ 9) - P(≤ 4)

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
    • straight P left parenthesis X less than x right parenthesis equals straight P left parenthesis X less or equal than x minus 1 right parenthesis and straight P left parenthesis X greater than x right parenthesis equals straight P left parenthesis X greater or equal than x plus 1 right parenthesis
    • For example: P(X < 5) = P(X ≤ 4) and P(X > 5) = P(X ≥ 6)
  • It helps to think about the range of integers you want
    • Identify the smallest and biggest integers in the range
  • If your range has no minimum or maximum then use 0 or n
    • straight P left parenthesis X less or equal than b right parenthesis equals straight P left parenthesis 0 less or equal than X less or equal than b right parenthesis
    • straight P left parenthesis X greater or equal than a right parenthesis equals straight P left parenthesis a less or equal than X less or equal than n right parenthesis
  • straight P left parenthesis a less than X less or equal than b right parenthesis equals straight P left parenthesis a plus 1 less or equal than X less or equal than b right parenthesis
    • P(5 < X ≤ 9) = P(6 ≤ X ≤ 9)
  • straight P left parenthesis a less or equal than X less than b right parenthesis equals straight P left parenthesis a less or equal than X less or equal than b minus 1 right parenthesis
    • P(5 ≤ X < 9) = P(5 ≤ X ≤ 8)
  • straight P left parenthesis a less than X less than b right parenthesis equals straight P left parenthesis a plus 1 less or equal than X less or equal than b minus 1 right parenthesis
    • P(5 < X < 9) = P(6 ≤ X ≤ 8)

Examiner Tip

  • If the question is in context then write down the inequality as well as the final answer
    • This means you still might gain a mark even if you accidentally type the wrong numbers into your GDC

Worked example

The random variable X tilde straight B left parenthesis 40 comma space 0.35 right parenthesis. Find:

i)
straight P left parenthesis X equals 10 right parenthesis.

4-5-2-ib-ai-aa-sl-binomial-prob-a-we-solution

ii)
straight P left parenthesis X less or equal than 10 right parenthesis.

4-5-2-ib-ai-aa-sl-binomial-prob-b-we-solution

iii)
straight P left parenthesis 8 less than X less than 15 right parenthesis.

4-5-2-ib-ai-aa-sl-binomial-prob-c-we-solution

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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.