A.P. STATISTICS LESSON 6.3 ( DAY 2 ) GENERAL PROBABILITY RULES ( EXTENDED MULTIPLICATION RULES )

ESSENTIAL QUESTION: How are the probabilities found that involve and or all combinations? Objectives:  To define the term intersection when referring to probability.  To use tree diagrams to find probabilities when finding extended multiplication problems.

Extended multiplication rules The union of a collection of events is the event that any of them occur. Intersection – 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. P(A and B and C) = P(A)P(B/A)P(C/A and B)

Tree diagrams 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.

Example 6.23  Page 372

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.

Bayes’s Rule If A and B are any events whose probabilities are not 0 or 1, P(A/B) = P(B/A)P(A) P(B/A)P(A) + P( B/A C )P(A C )

Independence The conditional probability P(B/A) 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 B occurs. If knowing that A occurs gives no additional information about B, then A and B are independent events.

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

Example 6.25 Page 376