Probabilities for Binomial Distributions (College Board AP® Statistics): Study Guide

Using the following two facts can speed up calculations:

• $\left(\begin{array}{c}n\\ 0\end{array}\right)=\left(\begin{array}{c}n\\ n\end{array}\right)=1$

• $\left(\begin{array}{c}n\\ 1\end{array}\right)=n$

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

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 $n=20$ and $p=0.5$, $P\left(X\le 7\right)=0.1315...$

• $\mathrm{binomcdf}\left(n=20,p=0.5,\mathrm{upper}=7\right)=0.1315...$

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.

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.

(a) Define the random variable of interest and state how the random variable is distributed.

(b) 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:

(a)

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 $n=10$ and $p=\frac{1}{30}$

(b)

The required probability is $P\left(X\ge 2\right)$

This is equal to $P\left(2\right)+P\left(3\right)+...+P\left(10\right)$

It is quicker to find the probability that a student is selected at most once and subtract this from 1, i.e. $1-\left(P\left(0\right)+P\left(1\right)\right)$

Find the probability of each outcome using $P\left(X=x\right)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{\left(1-p\right)}^{n-x}$

$\begin{array}{rcl}P\left(X\ge 2\right)& =& 1-P\left(X\le 1\right)\\ & =& 1-\left(P\left(0\right)+P\left(1\right)\right)\\ & =& 1-\left(\left(\begin{array}{c}10\\ 0\end{array}\right){\left(\frac{1}{30}\right)}^0{\left(1-\frac{1}{30}\right)}^{10-0}+\left(\begin{array}{c}10\\ 1\end{array}\right){\left(\frac{1}{30}\right)}^1{\left(1-\frac{1}{30}\right)}^{10-1}\right)\\ & =& 1-\left({\left(\frac{29}{30}\right)}^{10}+10\left(\frac{1}{30}\right){\left(\frac{29}{30}\right)}^9\right)\\ & =& 1-0.95815...\\ & =& 0.04184...\end{array}$

Check this on a calculator using $\mathrm{binomcdf}\left(n=10,p=\frac{1}{30},\mathrm{lower}=2,\mathrm{upper}=10\right)$

The probability that a particular student is selected to hand out the textbooks at least twice within 10 school days is 0.0418