Conditional Probability
and
Independence
Section 5.3
Reference Text:
The Practice of Statistics, Fourth Edition.
Starnes, Yates, Moore
Objectives
1. Conditional Probability
-
“what's the probability of Event B given that Event A has happened”
2. Independent Events
If the chances of event B occurring is not affected by whether event A
occurs, then A and B are independent!
3. Tree Diagrams!
Map out my probability calculations!
General Multiplication Rule
P(A ∩ B) = P(A) * P(B I A)
Special case of independent events, multiplication rule becomes
P(A ∩ B) = P(A) * P(B)
4. Conditional Probability formula
Conditional Probability
• The probability that one event happens given
that another event is already known to have
happened is called Conditional Probability.
• Suppose we known that event A has happened.
Then the probability that event B happens given
that event A has happened is denoted by
P(B I A)
Lets look at the male and pierced ears example!
Two way Tables
• Students in college stats class wanted to find out
how common it is for young adults to have their
ears pierced. They recorded data on two
variables- gender and whether the student had a
pierced ear – for all 178 people in class. The two
way table below displays the data.
Pierced ears?
Gender
Yes
No
Total
Male
19
71
90
Female
84
4
88
Total
103
75
178
A= male B= pierced ears
P(A) = P(male) = 90/178
P(B) = P(pierced ears) = 103/178
P(A ∩ B) = P(male and pierced ears) = 19/178
P(A U B) = P(male or pierced ears) = 174/178
Now lets turn out attention to some other interesting
probability questions…
Heads Up
• You might be thinking to yourself in the
previous slide and this next slide, “hey,
we’ve done this before! Is there a
connection between conditional probability
and the conditional distributions of Chapter
1?!”
• Of course! We have been doing probability
and conditional probability from the start!
Conditional Probability
• Conditional probability narrows your focus
in onto a specific event. Given a certain
condition….
P(A I B)
P( B I A)
Flashback!
• In 1912 the luxury liner Titanic, on its first voyage across
the Atlantic, struck an iceberg and sank. Some
passengers got off the ship in lifeboats, but many died.
The two-way table below gives information about adult
passengers who lived and who died, by class of travel.
Survival Status
Class of Travel
Survived
Died
First Class
197
122
Second Class
94
167
Third Class
151
476
Total
Total
Conditional Probability
•
•
•
•
P(survived I first class)
P(survived)
P(died I third class)
P(first class I survived)
Checking for Independence
• Sometimes applying a conditional probability
doesn’t affect the outcome and we get the same
probability as if we were just calculating the
probability of the event without the condition.
• When this happens, this is known as
Independent events- if the occurrence of one
event has no effect on the chance that the other
event will happen.
P(A I B) = 19/100
P(A) = 19/100
Checking for Independence
• In order to know if two events are independent
we need to compute probabilities.
• Example: Is there a relationship between gender
and handedness? To find out we used
CensusAtSchool’s Random Data Selector to
chose an SRS of 50 Australian high school
students who completed a survey. The two way
table displays data on the gender and dominant
hand of each student.
Checking for Independence
• To check whether the two events are independent, we
need to see whether knowing that one event has
happened affects the probability that the other event
occurs.
Dominant Hand
Gender
Right
Left
Total
Male
20
3
23
Female
23
4
27
Total
43
7
50
Are the events “male” and “left
handed” independent?
• Suppose we are told that the chosen student is male.
From the two way table
P(Left handed I male) = 3/23 =0.13
• The unconditional probability
P(left handed) = 7/50 =0.14
These two probabilities are close, but they’re not equal. So
the events “male” and “left handed” are not independent.
Tree Diagrams
General Multiplication Rule
• When I say to you, “what's the probability
of flipping a heads and heads again?”
• The word “and” implies multiplication!
– This same strategy works for trials that are not
independent as well!
• We can diagram the outcomes and probability of each
through a tree diagram!
Tree Diagram:
flipping a coin twice
General Multiplication Rule
• When we calculate the probability of
flipping two heads, we are following the
general multiplication rule!
P(A ∩ B) = P(A) * P(B I A)
Read as: “The Probability of A intersect B equals
the probability of flipping a heads multiplied by the
probability of flipping a second heads given that
the first flip was heads.”
Teens with online profiles
• The Pew Internet and American Life
Project finds that 93% of teenagers (ages
12 to 17) use the internet, and that 55% of
online teens have posted a profile on a
social-networking site
Question: What percent of teens are online
and have posted a profile? Create a tree
diagram. Lets do this on the board!
Check for Understanding
• A computer company makes desktops and laptop
computers at factories in three states- California, Texas,
and New York. The California factory produces 40% of
the company’s computers, the Texas factory makes
25%, and the remaining 35% are manufactured in New
York. Of the computers made in California, 75% are
laptops. Of those made in Texas and New York, 70%
and 50% respectively, are laptops. All computers are first
shipped to a distribution center in Missouri before being
sent out to stores. Suppose we select a computer at
random from the distribution center.
1. Construct a tree diagram to represent the situation.
2. Find the probability that it’s a laptop from California.
Show your work.
Independence: A Special
Multiplication Rule
• If A and B are independent events, then
the probability that A and B occur is
P(A ∩ B) = P(A) * P(B)
Note that this rule only applies to independent
events.
Lets look at an example.
The Challenger Disaster
• On January 28, 1986, Space Shuttle Challenger
exploded on takeoff. All seven crew members were
killed. Following the disaster, scientists and statisticians
helped analyze what went wrong. They determined that
the failure of O-ring joints in the shuttle’s booster rockets
was to blame. Under cold conditions that day, experts
estimated that the probability that an individual O-ring
joint would function properly was 0.977. But there were
six of these O-ring joints, and all six had to function
properly for the shuttle to launch safely.
• Assuming that O-ring joints succeed or fail
independently, find the probability that the shuttle would
launch safely under similar conditions.
The Challenger Disaster
• Assuming that O-ring joints succeed or fail
independently, find the probability that the shuttle would
launch safely under similar conditions.
• P(joint 1 OK and joint 2 OK and joint 3 OK and joint 4 OK and joint 5
OK and joint 6 OK) =
• (0.977) (0.977) (0.977) (0.977) (0.977) (0.977)= 0.87
• There is an 87% chance that the shuttle would launch
safely under similar conditions (and a 13% chance that it
wouldn’t).
Conditional Probability Formula
• If we rearrange the terms in the general
multiplication rule, we can get the formula
for the conditional probability P(A I B). We
start with the general multiplication rule:
P(A ∩ B) = P(A) * P(B I A)
Example
Cell Phone
No Cell Phone
Total
Landline
0.60
0.18
0.78
No Landline
0.20
0.02
0.22
Total
0.80
0.20
1.00
Objectives
1. Conditional Probability
-
“what's the probability of Event B given that Event A has happened”
2. Independent Events
If the chances of event B occurring is not affected by whether event A
occurs, then A and B are independent!
3. Tree Diagrams!
Map out my probability calculations!
General Multiplication Rule
P(A ∩ B) = P(A) * P(B I A)
Special case of independent events, multiplication rule becomes
P(A ∩ B) = P(A) * P(B)
4. Conditional Probability formula
Homework
Worksheet