1. Section 6.4: Conditional Probability

2. Conditional Probability of an Event Let E and F be two events with P(F) > 0. Conditional Probability is denoted by: P(E│F)

3. Example Would being criticized make one more likely to criticize others? Data from this study are below: Has been criticized Has not been criticized Total Has Criticized others 19827 Has not criticized others 23143166 Total42151193

4. Suppose that a chat room users is randomly selected. C = event that the individual has criticized others O = event that the individual has been personally criticized by others

6. Let’s Answer the Question How likely is it that someone being criticized would criticize others?

7. We can use the same information to answer the following question. Find the probability that an individual has been criticized by others given that he or she has criticized someone else.

8. Section 6.5: Independence

9. Two events are E and F are said to be Independent if P(E│F)=P(E) If E and F are not independent, they are said to be dependent events. If P(E│F) = P(E), it is also true that P(F│E)= P(F)

10. Multiplication Rule for Two Independent Events The events E and F are independent if and only if P(E ∩ F) = P(E)P(F) That is, independence implies the relation P(E ∩ F) = P(E)P(F), and this relation implies independence

11. Example: Hitchhiker’s Thumb The ability to bend the last joint of the thumb back at an angle of 60 degrees or more is determined by two random events: the mother and father. Both have to come to agreeing votes in order to have the gene. Suppose that there is a 0.10 probability that a parent contributes a positive hitchhiker’s thumb gene.

12. Because the events are independent, the probability that each parent contributes a positive hitchhiker’s thumb gene is, H+: P(mother contributes H+ ∩ father contributes H+) =P(mother contributes)P(father contributes) =(.10)(.10) =.01

13. Sampling with or without replacement Sampling is with replacement if, once selected, an individual or object is put back into the population before the next selection. Sampling is without replacement if, once selected, an individual or object is not returned to the population before subsequent selections.