Ch. 11 Conditional Probability
• P(A|B) = P(A and B)/ P(B),
assuming P(B) > 0
• P(B|A) = P(A and B)/ P(A), P(A) > 0
Examples
• In a monthly report, the local animal shelter
states that it currently has 24 dogs and 18 cats
available fop adoption. Eight of the dogs and
six of the cats are male. Find each of the
following probabilities.
• a) The pet is male given that it is a cat
• b) The pet is a cat, given that it is female
• c) The pet is dog given that it is a female
Example
• You draw a card at random from a standard
deck of 52 cards. Find the following
probabilities.
• a) The card is a heart, given that it is red
• b) The card is red, given that it is a heart
• c) The card is an ace, given that it is red
• d) The card is a queen, given that it is face
card
Ch. 11 Conditional Probability
Ch. 14/15 From randomness to Probability
Example
• 3. The Masterfoods Company says that before the
introduction of purple, yellow candies made up of 20% of
their plain M&M’s, red another 20% and orange, blue and
green each made up 10%. The rest were brown.
•
•
•
•
a) If you pick an M&M at random, what’s the probability that
i) it is brown
ii) it is yellow or orange?
iii) it is not green?
Examples
• 4. If you pick three M&M’s in a row, what s the
probability that
• a) they are all brown?
• b) the third one is the first one that is red?
• c) none are yellow?
• d) at least one is green
Example
• The probabilities that
an adult American man
has high blood pressure
and/or high cholesterol
are shown in the table High
Cholesterol
OK
HIGH
Blood
Pressure
OK
0.11
0.21
0.16
0.52
Example
• a) What is the probability that a man has both
conditions?
• b) What is the probability that a man has high blood
pressure?
• c) What is the probability that a man with high blood
pressure has high cholesterol?
• d) What is the probability that a man has high blood
pressure if it is known he has high cholesterol?
• e) Are high blood pressure and high cholesterol
independent?
Example
• A poll conducted by the
D
University of Montana
classified respondents
by gender and political
Male
36
party as shown in the
table. Is party affiliation
independent of gender?
Female 48
R
I
45 24
33 16
Example
D
R
I
M
.18
.22
.12
.52
F
.24
.16
.08
.48
.42
.38
.20
1
Chapter 11 Conditional Probability and Tree Diagram
1% of employees in a company are drug
users. A test was developed to identify the
drug users. If a person is a drug user, then
the test returns positive result 98% of the
time. If a person is not a drug user, the test
returns negative result 97% of the time.
Will you recommend this test to be
administered at your company?
Conditional Probability and Tree Diagrams
Leah is flying from Boston to Denver with a connection in
Chicago. The probability her first flight leaves on time is
0.15. If the flight is on time, the probability her luggage
will make the connection flight in Chicago is 0.95, but if the
first flight is delayed, the probability that the luggage will
make it is only 0.65.
a) What is the probability that her luggage arrives on time?
Answer: 0.695
b) Are the first flight leaving on time and the luggage making the
connection independent events? NO
Conditional Probability and Tree Diagrams
A package will be picked up by one of 3 delivery
vans, depending on which is nearest. The choices
are equal for any of the vans to make the pick up.
Past record for each van’s success in delivering
packages are given as follows:
• Prob(on time delivery for van 1) = .95
• Prob(on time delivery for van 2) =.96
• Prob(on time delivery for van 3) = .90.
• What is the probability a package will be delivered on time?
ANS 0.94
• Is package deliver independent of delivery van? NO
• If package is delivered on time, what is the probability that it
was delivered by Van 1? ANS 0.338
Bayes’ Theorem-Reversing the Conditioning
P ( B | A) 
P( A | B)P(B)
c
c
P( A | B)P(B)  P( A | B )P(B )
At a gas station, 40% of customers pump regular
gas, 35% pump midgrade gas, and 25% pump
premium grade gas. Of those who pump regular,
30% pay at least $30. Of those who pump midgrade,
50% pay at least $30. And of those who pump
premium, 60% pay at least $30.
a) What is the probability that the next customer
will pay more than $30? ANS 0.445
b) If the next customer pays more than $30, what is
the probability that he pumped regular gas?
ANS 0.269
Reversing the Condition
A firm that specializes in filing out income
tax forms has two types of accounts
• Large(gross income greater than $50000)
• Small(gross income less than $50000)
They find that 5% of the large accounts and
2% of the small accounts are audited by the
IRS. If 40% of their accounts are large
accounts, and an account is audited, what is
the probability that it is large? ANS 0.625