### 5.4

```Conditional Probability
●
For “or” probabilities
 The Addition Rule applies to two disjoint events
… the “easy” case
 The General Addition Rule applies to any two
events
●
For “and” probabilities
 The Multiplication Rule applies to two
independent events … the “easy” case
 The General Multiplication Rule, this section,
applies to any two events
●
Example
 Choosing cards from a deck of cards
 E = we chose a diamond as the first card
 We did not replace our first card
 F = we chose a heart as the second card
●
The probability of F happening, given that E
 There are 51 cards remaining
 13 of them are hearts




13/51 is called a conditional probability
The probability of choosing a heart is 13/52
The probability of choosing a heart, given
13/51
This can be written
P(Diamond | Heart) = 13/51


The notation for conditional probability
P(F|E)
is the probability of F given event E
Only the outcomes contained in the event E
are included in computing conditional
probabilities
●
●
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A group of adults are as per the following
table
Male
Female
Total
Right
Handed
38
42
80
Left
Handed
12
8
20
Total
50
50
100
We choose a person at random out of this
group
If E = “male” and F = “left handed”, compute
P(F) and P(F|E)
F = “left handed” … P(F) = 20/100 = 0. 20
● E = “male” … P(F|E) = probability of left
●
handed, given male = 12/50 = 0.24
 There are 50 males and 12 of them are left handed
 The probability of left handed, given male, is
12/50
●
●
The Conditional Probability Rule is
An interpretation of this is that we only
consider the cases when E occurs (i.e. P(E)),
and out of those, we consider the cases when
F occurs (i.e. P(E and F), since E always has to
occur)

We can take the Conditional Probability Rule
and rearrange it to be

This is the General Multiplication Rule
●
●
Example
For a student in a statistics class
 E = “did not do the homework” with P(E) = 0.2
 F = “the professor asks that student a question
about the homework” with P(F|E) = .9
What is the probability that the student did
not do the homework and the professor
homework?
P(E and F) = P(E) • P(F|E) = 0.2 • 0.9 = 0.18
●


Conditional probabilities P(F|E) represent
the chance that F occurs, given that E
occurs also
The General Multiplication Rule applies to
“and” problems for all events and involves
conditional probabilities


Suppose a single card is selected from a
standard 52- card deck. What is the
probability that the card drawn is a king?
Now suppose a single card is drawn from a
standard 52- card deck, but we are told that
the card is a heart. What is the probability
that the card drawn is a king?

According to the U. S. National Center for
Health Statistics, in 2002, 0.2% of deaths in
the United States were 25- to 34- year- olds
whose cause of death was cancer. In addition,
1.97% of all those who died were 25 to 34
years old. What is the probability that a
randomly selected death is the result of
cancer if the individual is known to have been
25 to 34 years old?

According to the U. S Census Bureau, 19.1%
of U. S. households are in the Northeast. In
addition, 4.4% of U. S. households earn
\$75,000 per year or more and are located in
the Northeast. Determine the probability that
a randomly selected U. S. household earns
more than \$ 75,000 per year, given that the
household is located in the Northeast.
Died from Cancer
Did not die from
Cancer
Never Smoked Cigars
782
120474
Former Cigar Smoker
91
7757
Current Cigar Smoker
141
7725
( a) What is the probability that a
randomly selected individual from the
study who died from cancer was a
former cigar smoker?
( b) What is the probability that
a randomly selected individual
from the study who was a
former cigar smoker died from
cancer?
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
A bag of 30 tulip bulbs purchased from a
nursery contains 12 red tulip bulbs, 10 yellow
tulip bulbs, and 8 purple tulip bulbs.
( a) What is the probability that two randomly
selected tulip bulbs are both red?
( b) What is the probability that the first bulb
selected is red and the second yellow?
( c) What is the probability that the first bulb
selected is yellow and the second is red?



Due to a manufacturing error, three cans of
regular soda were accidentally filled with diet
soda and placed into a 12- pack. Suppose
that two cans are randomly selected from the
case.
( a) Determine the probability that both
contain diet soda.
( b) Determine the probability that both
contain regular soda. Would this be unusual?
```