1. General Probability Rules
AP STATISTICS
Section 6.3
General Probability Rules

2. What We Already Know

3. What We Already Know Cont…

4. Events that are not Disjoint

5. Example – Prosperity and Education
Call a household prosperous if its income exceeds $100,000. Call the household educated if the householder completed college. Select an American household at random, and let A be the event that the selected household is prosperous and B the event that it is educated. According to the Census Bureau, P(A) = 0.134, P(B) = 0.254, and the joint probability that a household is both prosperous and educated is P(A and B) =
What is the probability P(A or B) that the household selected is either prosperous or educated?
P(A or B) = P(A) + P(B) – P(A and B) = – = 0.308

6. Venn Diagram – Prosperity and Education
Draw a Venn diagram that shows the relation between events A and B. Indicate each of the events listed below on your diagram and use the information from the previous problem to calculate the probability of each event. Finally, describe in words what each event is.
A and B
A and Bc
Ac and B
Ac and Bc

7. Solution
Venn Diagram

8. Solution Cont… Household is both prosperous and educated;
P(A and B) = 0.080
Household is prosperous but not educated;
P(A and Bc) = P(A) – P(A and B)
= – = 0.054
Household is not prosperous but is educated;
P(Ac and B) = P(B) – P(A and B)
= – = 0.174
Household is neither prosperous nor educated;
P(Ac and Bc) = 1 – = 0.692

9. Conditional Probability
The probability we assign to an event can change if we know that some other event has occurred.
Slim is a professional poker player. He is dealt a hand of four cards. Find the probability that he gets an ace.
P(ace) =
Find the conditional probability that slim gets another ace given he has an ace already in his hand.
P(ace | 1 ace in 4 visible cards) =

10. Conditional Probability Cont…
The notation P(B|A) is a conditional probability and means the probability of B given that A has occurred.

11. Example – Municipal Waste
Municipal Waste Collected in the U.S. (in millions of tons)
Paper
Aluminum
Glass
Plastic
Other
Total
Recycled
26.5
1.1
3.0
0.7
13.7
Not Recycled
51.3
1.9
10.7
19.3
78.7
Use the information shown in the table to find each probability.
P(Recycled)
P(Paper)
P(Recycled and Paper)

12. Solution P(Recycled) = 45/206.9 = 0.22 P(Paper) = 77.8/206.9 = 0.38
P(Recycled and Paper) = 26.5/206.9 = 0.13

13. Conditional Probability Rule
Try:
P(Paper | Recycled)
P(Recycled | Aluminum)
P(Aluminum | Recycled)

14. Conditional Probability with Tree Diagrams
Only 5% of male high school basketball, baseball, and football players go on to play at the college level. Of these, only 1.7% enter major league professional sports. Sometimes gifted athletes are able to skip college and play professionally right out of high school, but this only occurs about 0.01% of the time.
P(high school athlete goes pro)
P(pro athlete played at the college level)
P(pro athlete didn’t play at the college level)