40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
The Binomial
Distribution
ST-L5 Objectives:
To solve problems using Binomial Probabilities.
Learning Outcome B-4
Slide 1
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
Normal Distributions deal with Continuous Data, and typically analyze
how individual observations deviate from central values.
Binomial Distributions deal with Discrete Data. A binomial experiment
has the following properties
•The experiment consists of n repeated trials.
•Each trial can result in just two possible outcomes. We call one of these
outcomes a success and the other, a failure.
•The probability of success, denoted by p, is the same on every trial.
•The trials are independent; that is, the outcome on one trial does not
affect the outcome on other trials.
Theory – Characteristics of a
Binomial Distribution
Slide 2
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
Consider the following statistical experiment.
You flip a coin 32 times and count the number of times the coin lands on
heads.
This is a binomial experiment because:
•The experiment consists of repeated trials. We flip a coin 32 times.
•Each trial can result in just two possible outcomes - heads or tails.
•The probability of success is constant - 0.5 on every trial.
•The trials are independent; that is, getting heads on one trial does not affect
whether we get heads on other trials.
Example – Identifying a Binomial
Distribution
Slide 3
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
When dealing with Binomial Distributions, we attend to a number of
parameters
n = the number of trials
p = the probability of success
q = the probability of failure = 1 - p
x = the number of successes in n trials, and 0 x n
P(x) = probability of getting exactly x successes in n trials
{The value of P(x) is the solution that we are after}
Note that the value 'x' always represents the same kind of outcome as 'p'.
Theory - Notation Used in Binomial Problems
Slide 4
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
A manufacturer produces 24 yard alarms per week. Six percent of all the
alarms produced are defective. What is the probability of getting two
defective alarms in one week?
First things first: Identify your parameters
n = the number of trials
p = the probability of success
q = the probability of failure = 1 - p
x = the number of successes in n trials, and 0 x n
P(x) = probability of getting exactly x successes in n trials
Theory - Using IT to Find a Binomial Probability
n = 24
p = 0.06
q = 0.94
x=2
P(x) = ?
Slide 5
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
A manufacturer produces 24 yard alarms per week. Six percent of all the
alarms produced are defective. What is the probability of getting two
defective alarms in one week?
Solution using Winstats.
1. Select Windows > Probability > Binomial… A
new window should open.
2. In the new window Edit > Parameters. A new
small window should open.
3. In the parameters window, enter: trials = 24
prob = 0.06
4. Select Calc > Table in the Binom[ window. A new
window showing a probability table should open.
You can use this table to answer any binomial
probability question.
Theory - Using IT to Find a Binomial Probability
Slide 6
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
As shown on the table, the probability of having two faulty alarm systems is
0.255,
i.e., P(x = 2) = 0.25470.
This means that there is a 25.5 percent chance of having two faulty alarm
systems in one week.
Theory - Using IT to Find a Binomial Probability
Slide 7
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
Sample Binomial Problem (1)
Slide 8
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
There are five children in a family. Assume that boys and girls are equally likely.
a. What is the probability that three are girls?
b. What is the probability that there are at most three girls (i.e., there may be 0,
1, 2, or 3 girls)?
c. What is the probability that there are more than two girls?
Using IT to Find a Range of Binomial
Probabilities
Slide 9
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
Sample Binomial Problem (2)
Slide 10
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
End Day 1
do q. 1-7
Sample Binomial Problem (3)
Slide 11
40S Applied Math
Mr. Knight – Killarney School
The histograms of (some of) the
binomial distributions on the
previous pages looked similar to
normal distributions. Therefore, it
is fair to conclude that, in certain
cases, a binomial distribution can
be reasonable approximated by a
normal distribution.
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
Normal
Not Normal
Normal Approximation to the Binomial
Distribution
Slide 12
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
Recall:
In a binomial distribution, we
used values for n and p to solve
problems, where:
n = number of trials, and
p = probability of success
In a normal distribution, we used
values for m and s to solve
problems, where:
m = the population mean, and
s = the standard deviation
Normal Approximation to the Binomial
Distribution
Slide 13
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
If certain conditions exist, a binomial distribution will be
approximately normal.
Check to see that
np  5 and nq  5
If these conditions exist, then the binomial distribution can be
described by a normal distribution which has values of
m = np
and
s=
Normal Approximation to the Binomial
Distribution (cont’d)
npq
npq
Slide 14
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
Border patrol officers estimate that 10 percent of the vehicles
crossing the US - Canada border carry undeclared goods. One day
the officers searched 350 randomly selected vehicles. What is the
probability that 40 or more vehicles carried undeclared goods?
Solution:
First check for the normal approximation to a binomial distribution.
np = (350)(0.10) = 35
nq = (350)(0.90) = 315
Therefore, the approximation is reasonably close to a normal
distribution because:
np  5 and nq  5
Example: Solve a Binomial Distribution
Problem as a Normal Distribution Problem
Slide 15
40S Applied Math
Mr. Knight – Killarney School
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
Then, calculate the mean and standard deviation, and solve using
your normal distribution techniques:
P(x  40) = 0.186
Therefore, there is about an 18.6 percent chance that 40 or more
vehicles were carrying undeclared goods.
Example: Solve a Binomial Distribution Problem
as a Normal Distribution Problem (cont’d)
Slide 16
40S Applied Math
Mr. Knight – Killarney School
Sample Problem: Solve as a Binomial
Distribution and Normal Distribution
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
Slide 17
40S Applied Math
Mr. Knight – Killarney School
Sample Problem: Solve as a Binomial
Distribution and Normal Distribution
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
Slide 18
40S Applied Math
Mr. Knight – Killarney School
Sample Problem: Solve as a Binomial
Distribution and Normal Distribution
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
Slide 19
40S Applied Math
Mr. Knight – Killarney School
Sample Problem: Solve as a Binomial
Distribution and Normal Distribution
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
Slide 20
40S Applied Math
Mr. Knight – Killarney School
Sample Problem: Solve a Binomial Problem as a
Normal Distribution Problem
Unit: Statistics
Lesson: ST-5 The Binomial Distribution
Slide 21