AP®StatisticsCollege BoardRevision NotesUnit 4: Probability, Random Variables & Probability DistributionsBinomial & Geometric DistributionsProbabilities for Binomial Distributions

Probabilities for Binomial Distributions (College Board AP® Statistics)

Revision Note

Dan Finlay

Expertise

Maths Lead

Probabilities for binomial distributions

How do I calculate the probability of a single outcome using a binomial distribution?

  • Let X be a discrete random variable following a binomial distribution with parameters n and p

    • X tilde B open parentheses n comma space p close parentheses

  • The formula for P open parentheses X equals x close parentheses, where x equals 0 comma space 1 comma space 2 comma space... comma space n, is

    • P open parentheses X equals x close parentheses equals open parentheses table row n row x end table close parentheses space p to the power of x open parentheses 1 minus p close parentheses to the power of n minus x end exponent

      • This is given in the exam

    • where open parentheses table row n row x end table close parentheses equals C presubscript n subscript x equals fraction numerator n factorial over denominator x factorial open parentheses n minus x close parentheses factorial end fraction

      • n factorial equals n times open parentheses n minus 1 close parentheses times open parentheses n minus 2 close parentheses times... times 2 times 1

      • 0 factorial equals 1

    • e.g. if X tilde B open parentheses 20 comma space 0.3 close parentheses then P open parentheses X equals 5 close parentheses equals open parentheses table row 20 row 5 end table close parentheses open parentheses 0.3 close parentheses to the power of 5 open parentheses 1 minus 0.3 close parentheses to the power of 20 minus 5 end exponent

      • this can be written as P open parentheses X equals 5 close parentheses equals fraction numerator 20 factorial over denominator 5 factorial times 15 factorial end fraction open parentheses 0.3 close parentheses to the power of 5 open parentheses 0.7 close parentheses to the power of 15

Exam Tip

Using the following two facts can speed up calculations:

  • open parentheses table row n row 0 end table close parentheses equals open parentheses table row n row n end table close parentheses equals 1

  • open parentheses table row n row 1 end table close parentheses equals n

these are true for any non-negative integer value of n.

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. P open parentheses 2 less or equal than X less than 6 close parentheses equals P open parentheses 2 close parentheses plus P open parentheses 3 close parentheses plus P open parentheses 4 close parentheses plus P open parentheses 5 close parentheses

  • 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. P open parentheses X greater than 3 close parentheses equals 1 minus open parentheses P open parentheses 0 close parentheses plus P open parentheses 1 close parentheses plus P open parentheses 2 close parentheses plus P open parentheses 3 close parentheses close parentheses

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 P open parentheses X equals x close parentheses

    • This might be shown as BPD, Binomial PD or binompdf

    • You need to enter:

      • the value of n

      • the value of p

      • the value of the outcome x

  • The binomial cumulative distribution function finds the probability of an event containing outcomes within an interval P open parentheses lower less or equal than X less or equal than upper close parentheses

    • This might be shown as BCD, Binomial CD or binomcdf

    • You need to enter:

      • the value of n

      • the value of p

      • 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 binompdf open parentheses n comma space p comma space x close parentheses and binomcdf open parentheses n comma space p comma space lower comma space upper close parentheses

    • e.g. for Casio models, use binomialpd open parentheses x comma space n comma space p close parentheses and binomialcd open parentheses lower comma space upper comma space n comma space p close parentheses

Exam Tip

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:

  • X has a binomial distribution with n equals 20 and p equals 0.5, P open parentheses X less or equal than 7 close parentheses equals 0.1315...

  • binomcdf open parentheses n equals 20 comma space p equals 0.5 comma space upper equals 7 close parentheses equals 0.1315...

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, X, 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

X has a binomial distribution with n equals 10 and p equals 1 over 30

(ii) The required probability is P open parentheses X greater or equal than 2 close parentheses

This is equal to P open parentheses 2 close parentheses plus P open parentheses 3 close parentheses plus... plus P open parentheses 10 close parentheses

It is quicker to find the probability that a student is selected at most once and subtract this from 1, i.e. 1 minus open parentheses P open parentheses 0 close parentheses plus P open parentheses 1 close parentheses close parentheses

Find the probability of each outcome using P open parentheses X equals x close parentheses equals open parentheses table row n row x end table close parentheses space p to the power of x open parentheses 1 minus p close parentheses to the power of n minus x end exponent

table row cell P open parentheses X greater or equal than 2 close parentheses end cell equals cell 1 minus P open parentheses X less or equal than 1 close parentheses end cell row blank equals cell 1 minus open parentheses P open parentheses 0 close parentheses plus P open parentheses 1 close parentheses close parentheses end cell row blank equals cell 1 minus open parentheses open parentheses table row 10 row 0 end table close parentheses open parentheses 1 over 30 close parentheses to the power of 0 open parentheses 1 minus 1 over 30 close parentheses to the power of 10 minus 0 end exponent plus open parentheses table row 10 row 1 end table close parentheses open parentheses 1 over 30 close parentheses to the power of 1 open parentheses 1 minus 1 over 30 close parentheses to the power of 10 minus 1 end exponent close parentheses end cell row blank equals cell 1 minus open parentheses open parentheses 29 over 30 close parentheses to the power of 10 plus 10 open parentheses 1 over 30 close parentheses open parentheses 29 over 30 close parentheses to the power of 9 close parentheses end cell row blank equals cell 1 minus 0.95815... end cell row blank equals cell 0.04184... end cell end table

Check this on a calculator using binompcdf open parentheses n equals 10 comma space p equals 1 over 30 comma space lower equals 2 comma space upper equals 10 close parentheses

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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    Dan Finlay

    Author: Dan Finlay

    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.