1.1.2 Calculating Binomial Probabilities

Throughout this section we will use the random variable . 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
  •  can be 5, 6, 7, 8, 9 10, 12, 15, 20, 25, 30, 40, 50
  •  can be 05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5
  •  can be different values depending on n
• The probabilities are rounded to 4 decimal places
• The values of only go up to 0.5
  • You can instead count the number of failures  if the probability of success is bigger than 0.5
  • Remember , which leads to identities:
    • 
    • 
    • 

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
  
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    • The number of times this can happen is calculated by the binomial coefficient 
• You can also use the tables for the Binomial Cumulative Distribution Function in the formulae booklet
  
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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 
  • 

How do I find P(X ≥ 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
• 
• As x  is an integer then as the probability of X is zero for non-integer values for a binomial distribution
• Therefore, to calculate :
  • 
  • For example: 

How do I find  P(a ≤ X ≤ 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
• 
• As X is an integer then  as the  for non-integer value of x for a binomial distribution
• Therefore to calculate :
  •  
  • For example: 

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
  • and 
  • For example:  and 
  • 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, : 
  • You want the integers 5 to 10
  • You want the integers up to 10 excluding the integers up to 4
  • 
• 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

Exam Tip