Probabilities for Binomial Distributions (College Board AP® Statistics)
Study Guide
Written by: Dan Finlay
Reviewed by: Lucy Kirkham
Probabilities for binomial distributions
How do I calculate the probability of a single outcome using a binomial distribution?
Let be a discrete random variable following a binomial distribution with parameters and
The formula for , where , is
This is given in the exam
where
e.g. if then
this can be written as
Examiner Tips and Tricks
Using the following two facts can speed up calculations:
these are true for any non-negative integer value of .
How do I calculate the probability of an event using a binomial distribution?
To calculate the probability of an event
find the probabilities of each outcome in the event
add the probabilities together
e.g.
To calculate the probability of an event using the complement of the event
find the probabilities of the outcomes that are not in the event
add the probabilities together
subtract from 1
e.g.
How can I calculate probabilities using a binomial distribution on a calculator?
Most graphical calculators can calculate probabilities for a binomial distribution
The binomial probability distribution function finds the probability of a single outcome
This might be shown as BPD, Binomial PD or binompdf
You need to enter:
the value of
the value of
the value of the outcome
The binomial cumulative distribution function finds the probability of an event containing outcomes within an interval
This might be shown as BCD, Binomial CD or binomcdf
You need to enter:
the value of
the value of
the value of the lower bound
the value of the upper bound
Some calculators do not have an option for the lower bound
in this case, the lower bound is 0
you may need to find two cumulative probabilities and subtract one from the other
Check your calculator's manual to see the syntax for these functions
e.g. for TI models, use and
e.g. for Casio models, use and
Examiner Tips and Tricks
You are allowed to use your calculator in the exam but you must clearly state:
the distribution including the parameters,
the event that you are finding the probability of.
For example, the following are acceptable:
has a binomial distribution with and ,
The safest approach is to calculate the probability using the formula and show all of your work. Then, you can check your answer on your calculator.
Worked Example
There are 30 students in a math class. Each day, the teacher randomly selects a student to hand out the textbooks. Each of the 30 students is equally likely to be selected each day and the same student could be selected more than once. Each day's selection is independent from every other day.
Consider the probability that a particular student is selected to hand out the textbooks at least twice within 10 school days.
(i) Define the random variable of interest and state how the random variable is distributed.
(ii) Determine the probability that a particular student is selected to hand out the textbooks at least twice within 10 school days. Show your work.
Answer:
(i) Let the random variable of interest, , represent the number of times a particular student is selected to hand out the textbooks within 10 school days
The conditions are met for a binomial distribution
has a binomial distribution with and
(ii) The required probability is
This is equal to
It is quicker to find the probability that a student is selected at most once and subtract this from 1, i.e.
Find the probability of each outcome using
Check this on a calculator using
The probability that a particular student is selected to hand out the textbooks at least twice within 10 school days is 0.0418
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