# 3.2.2 Calculating Binomial Probabilities

## 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.

#### Where does the formula for a binomial distribution come from?

• The formula for calculating an individual binomial probability is
• If there are r 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 r successes
•   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
• is 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  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  is the random variable . Find:

(i)
(ii)
(iii)
(iv)

#### 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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