1. Chapter 4, part D Download this file. Download this file.

2. V. B. Independent Events Two events are independent if the fact that one has already occurred has no effect on the probability that the second will occur. P(A  B) = P(A) Example: Given Bill Clinton has a cat named “Socks” has no effect on the probability that another American will name their dog “Socks”.

3. Back to our example. Is marital status independent of age in the nightclub sample? If it is, P(S|U) = P(S) P(S|U) = P(S  U)/P(U) =.55/.65 =.846 P(S) =.75, so P(S|U)  P(S) This tells us that a person is more likely to be single, if we know they’re under 30, than any one random person is likely to be single with no information about their age.

4. C. The Multiplication Law This law is used to find the intersection of two events, P(A  B). Recall: P(A|B) = P(A  B)/P(B) Rewrite this and you have the multiplication law. P(A  B) = P(A|B)P(B), or P(A  B) = P(B|A)P(A)

5. Example of the multiplication law. Suppose a daily newspaper has the following subscription information: D= customer subscribes to the daily paper P(D)=.84 S= customer has the Sunday subscription. P(S|D)=.75 P(S  D) = P(S|D)P(D)=.75(.84) =.63 This means if you pick a customer out of their database the probability that they subscribe to both the daily and the Sunday papers is.63.

6. D. Mutually exclusive vs. Independent events Mutually exclusive events are those that don’t share any common sample points. P(A  B)=0 Independent events are those where the occurrence of one event has no effect on the probability of the other occurring. P(A  B) = P(A)

7. An example. Customers at a nightclub can either be under 21 (U) or over 21 years old (O). P(O)=.70P(U)=.30 These are mutually exclusive events; you can’t be in both sets! P(U  O)=0. But they are dependent events.

8. How does this work? Given that a person is over 21, what is the probability that they are under 21? Zero!!!! P(U|O) = P(U  O)/P(O) = 0/.70 = 0  P(U) =.30 Thus mutually exclusive events are necessarily dependent events because if you are in one set you are, by definition, precluded from being in the other set.