4.2.2 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.

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 '' value - the value of  for which you want to find 
  • The '' value - the number of trials
  • The '' value - the probability of success
• Some calculators will give you the option of listing the probabilities for multiple values  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
  •  
    • If there are successes then there are  failures
    • The number of times this can happen is calculated by the binomial coefficient 
    • 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
    •   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 
  • Some calculators can find 
• 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 x  for which you want to find
    • Some will instead ask for lower and upper bounds
    • For this lower would be 0 and upper would be x
  • The '' value - the number of trials
  • The '' 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  for the lower bound and for the upper bound
  • Otherwise, you will need some extra identities
• : 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
• 
• 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) ?

• 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
• : 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
• 
• 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: 

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)