1. Conditional Probability
5.4
Conditional Probability

2. Recall For “or” probabilities For “and” 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

3. Example
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 has already happened, is 13/51
There are 51 cards remaining
13 of them are hearts

4. Conditional Probability
13/51 is called a conditional probability
The probability of choosing a heart is 13/52
The probability of choosing a heart, given that we had already chosen a diamond, is 13/51
This can be written
P(Diamond | Heart) = 13/51

5. Conditional Probability
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

6. Example A group of adults are as per the following table
We choose a person at random out of this group
If E = “male” and F = “left handed”, compute P(F) and P(F|E)
Male
Female
Total
Right Handed 38 42 80
Left Handed 12 8 20 50
100

7. 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

8. Conditional Probability Rule
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)

9. General Multiplication Rule
We can take the Conditional Probability Rule
and rearrange it to be
This is the General Multiplication Rule

10. Example 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 asks that student a question about the homework?
P(E and F) = P(E) • P(F|E) = 0.2 • 0.9 = 0.18

11. Summary
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

12. Examples
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?

13. Example
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?

14. Example
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.

15. Example
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?

16. Example
( b) What is the probability that a randomly selected individual from the study who was a former cigar smoker died from cancer?

17. Example
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?

18. Example
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?