1. Chapter 6 Section 6.3 Part 2 – General Probability Rules

2. Extended multiplication rules • Recall that the union of a collection of events is the event that any of them occur. • The intersection of any collection of events is the event that all of the events occur. • To extend the multiplication rule to the probability that all of several events occur, the key is to condition each event on the occurrence of all of the preceding events. • See example 6.22 on p.372

3. Tree Diagrams Revisited • Probability problems often require us to combine several of the basic rules into a more elaborate calculation. • Each segment in the tree is one stage of the problem. Each branch shows a path that must be taken to achieve the next branch. • The probability written on each segment is the conditional probability that that segment is given after reaching that point from each branch.

4. Example 6.23 • See example 6.23 on p.373

5. Tree diagrams cont. • The tree diagrams combine the addition and multiplication rules: • The multiplication rule says that the probability of reaching the end of any complete branch is the product of the probabilities written on its segments. • The probability of any outcome is then found by adding the probabilities of all branches that are part of that event.

6. Independence • The conditional probability is generally not equal to the unconditional probability P(B). • That is because the occurrence of event A generally gives us some additional information about whether or not event Boccurs. • If knowing that A occurs gives no additional information about B, then A and B are independent events.

7. Independent events • The formal definition states: • Two events A and B that both have positive probability are independent if: = P(B) • We now see that the multiplication rule for independent events , is a special case of the general multiplication rule,

8. Example 6.25 • See example 6.25 on p.376

9. Homework: p.378-381 #’s 63, 64, 68, 70-73, 75, & 77