General Probability Rules AP STATISTICS Section 6.3 General Probability Rules

What We Already Know

What We Already Know Cont…

Events that are not Disjoint

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) = 0.080. 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) = 0.134 + 0.254 – 0.080 = 0.308

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

Solution Venn Diagram

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) = 0.134 – 0.080 = 0.054 Household is not prosperous but is educated; P(Ac and B) = P(B) – P(A and B) = 0.254 – 0.080 = 0.174 Household is neither prosperous nor educated; P(Ac and Bc) = 1 – 0.308 = 0.692

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) =

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

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)

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

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

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)