Skip to main content

QUAN 2010 Notes: Introduction to Business Statistics

Omar Ortega

Section 5.3 Binomial Distribution

Definition 5.3.1.

A binomial experiment is a probability experiment possessing the following characteristics:
  1. It has a fixed number of trials, \(n\text{.}\)
  2. Each trial has only two possible outcomes: success or failure.
  3. The probability of success, \(p\text{,}\) and the probability of failure, \(q\text{,}\) are constant.
  4. Each trial is independent of the other trials.
The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution.

Exercise 5.3.2. Binomial Multiple Choice Problem.

    Which of the following are binomial experiments?
  • Taking a sample of 50 employees and counting how many of them have been at the company for at least 5 years.
  • Taking a standardized test 3 times and counting how many times I got a passing score.
  • Flipping a coin 100 times and counting how many times I got “tails”.
  • Taking a group of 100 students and finding their average GPA.
  • Tracking how long it took me to drive to work each day over a month.
The probability of exactly \(X\) successes in \(n\) trials with probability of success, \(p\text{,}\) is
\begin{equation*} P(X)=\frac{n!}{X!(n-X)!}p^X(1-p)^{n-X}\text{ for }X=0,1,2,...,n \end{equation*}
In this class, we can use Excel (and/or tables) instead of the formula above to calculate probabilities for a binomial distribution. (As the number of trials in a binomial experiment increases, calculating the probabilities using the complex formula becomes tedious.)
In Excel:
\(\text{The formula is } BINOM.DIST(x,n,p,\text{cumulative})\text{, where }\text{cumulative=TRUE ( or fewer successes) or FALSE (exactly successes)}\)
\begin{equation*} \boxed{ P(X\leq x)=BINOM.DIST(x,n,p,TRUE)\;\;\;\text{ and }\;\;\; P(X=x)=BINOM.DIST(x,n,p,FALSE) } \end{equation*}
We can also use tables to find probabilities. The appropriate table from your book is Table 1, and here is a link to a table you can use as well: https://uwf.edu/media/university-of-west-florida/colleges/cse/departments/mathematics-and-statistics/documents/student-resources/Binomial-Tables-1.pdf
Good news! The mean and standard deviation of a binomial distribution are simple computations:
\begin{equation*} \boxed{\mu=np}\;\;\;\;\text{ and }\;\;\;\; \boxed{\sigma=\sqrt{npq}} \end{equation*}

Exercise 5.3.3. In-Class Binomial Matching Activity.

In class we’re going to do a matching activity to practice calculating probabilities for binomial distributions. Below are the answers to this in-class activity.
Answer.
10 A
9 C
8 E
7 F
6 B
5 Q
4 Z
3 P
2 U
1 T
In Exercise 5.3.4,Exercise 5.3.5, Exercise 5.3.6, and Exercise 5.3.7, you can get a bit more practice calculating binomial probabilities.

Exercise 5.3.4.

Suppose a bank gives out a number of loans. Each of the \(6\) loans has a \(0.19\) chance of defaulting. Suppose that whether or not each loan default is independent of the other loans.
What is the probability that \(3\) loans default? Round your answer to four decimal places:

Exercise 5.3.5.

Suppose a bank gives out a number of loans. Each of the \(6\) loans has a \(0.19\) chance of defaulting. Suppose that whether or not each loan default is independent of the other loans.
What is the probability that \(4\) loans default? Round your answer to four decimal places:

Exercise 5.3.6.

Suppose a bank gives out a number of loans. Each of the \(6\) loans has a \(0.19\) chance of defaulting. Suppose that whether or not each loan default is independent of the other loans.
What is the probability that less than \(3\) loans default? Round your answer to four decimal places:

Exercise 5.3.7.

Suppose a bank gives out a number of loans. Each of the \(6\) loans has a \(0.19\) chance of defaulting. Suppose that whether or not each loan default is independent of the other loans.
What is the probability that between \(2\) and \(4\) (inclusive) loans default? Round your answer to four decimal places:
Now let’s get more practice working with other examples of binomial distributions.

Exercise 5.3.8.

(Donnelly 5.19)
According to Fortune, as of January 2018, \(5\%\) of chief executive officers (CEOs) were women. Answer the following questions based on a random sample of 12 CEOs

(a)

Why does this scenario fit a binomial experiment? Define the random variable, \(X\text{,}\) and clearly identify \(n\) and \(p\text{.}\)
Answer.
\(X=\) number of CEOs out of 12 that are women
\(n=12\text{:}\) the fixed number of trials; \(p=0.05\text{:}\) probability of “success” (fixed)
The 12 trials are independent.
There are only 2 outcomes: “success”: CEO is a woman; “failure”: CEO is not a woman

(b)

What is the probability that fewer than four CEOs were women?
Answer.
\begin{equation*} P(X\lt 4)=P(X\leq 3)=BINOM.DIST(3,12,0.05,TRUE)\approx 0.9978 \end{equation*}

(c)

What is the probability that at least two CEOs were women?
Answer.
\begin{equation*} P(X\geq 2)=1-P(X\leq 1)=1-BINOM.DIST(1,12,0.05,TRUE)\approx 0.1183 \end{equation*}

(d)

What is the probability that exactly one CEO was a woman?
Answer.
\begin{equation*} P(X=1)=BINOM.DIST(1,12,0.05,FALSE)\approx 0.3413 \end{equation*}

(e)

What are the mean and standard deviation number of women CEOs?
Answer.
  • \(\displaystyle \mu=np=(12)(0.05)=0.6\)
  • \(\displaystyle \sigma=\sqrt{npq}=\sqrt{(12)(0.05)(1-0.05)}\approx 0.755\)

Exercise 5.3.9.

(Donnelly 5.20)
An e-commerce website claims that \(7\%\) of people who visit the site make a purchase. Use Excel to answer the question below based on a random sample of 15 people who visited the website.
What is the probability that none of the people will make a purchase? (Round your answer to 4 decimal places.)

Exercise 5.3.10.

(Donnelly 5.20)
An e-commerce website claims that \(7\%\) of people who visit the site make a purchase. Use Excel to answer the question below based on a random sample of 15 people who visited the website.
What is the probability that less than 3 people will make a purchase? (Round your answer to 4 decimal places.)

Exercise 5.3.11.

(Donnelly 5.20)
An e-commerce website claims that \(7\%\) of people who visit the site make a purchase. Use Excel to answer the question below based on a random sample of 15 people who visited the website.
What is the probability that at least one person will make a purchase? (Round your answer to 4 decimal places.)

Exercise 5.3.12.

(Donnelly 5.20)
An e-commerce website claims that \(7\%\) of people who visit the site make a purchase. Use Excel to answer the question below based on a random sample of 15 people who visited the website.
Suppose that out of the 15 customers, 5 made a purchase. What conclusions can be drawn about the sample?
Answer.
\begin{equation*} P(X=5)=BINOM.DIST(5,15,0.07,FALSE)\approx 0.0024 \end{equation*}
Assuming \(7\%\) of customers make a purchase, the probability that 5 out of 15 make a purchase is very low (\(\lt 1\%\text{,}\) so very unlikely).
It is doubtful that the true percentage of customers who make a purchase is \(7\%\text{.}\)
PrevTopNext