PROBABILITY RULES AND TREES

1. PROBABILITY RULES AND TREES • Rule of complement • Addition rule • Multiplication rule • Probability tree

2. RULE OF COMPLEMENT • The simplest probability rule involves the complement of an event. • If the probability of A is P(A), then the probability of its complement, P(Ac), is P(Ac)=1- P(A) • Equivalently, the probability of an event and the probability of its complement sum to 1. P(A) + P(Ac)=1

3. RULE OF COMPLEMENT THE BENDRIX SITUATION • Find P(Bc) using the rule of complements • Does the rule of complements give the same result as it is given by the frequencies?

4. ADDITION RULEMUTUALLY EXCLUSIVE EVENTS • We say that events are mutually exclusive if at most one of them can occur. That is, if one of them occurs, then none of the others can occur. • Let A1 through An be any n mutually exclusive events. Then the addition rule of probability involves the probability that at least one of these events will occur. P(at least one of A1 through An) = P(A1) + P(A2) +  + P(An)

5. ADDITION RULE EXHAUSTIVE EVENTS • Events can also be exhaustive, which means that they exhaust all possibilities. Probabilities of exhaustive events add up to 1. • If A and B are exhaustive, P(A)+ P(B)=1 • If A,B and C are exhaustive, P(A)+ P(B)+ P(C)=1

6. ADDITION RULE THE BENDRIX SITUATION • Interpret the events E1 = (A and B) E2 = (A and BC)

7. ADDITION RULE THE BENDRIX SITUATION • Are the events E1 and E2 mutually exclusive? • Verify the following P(A) = P(E1)+P(E2)

8. ADDITION RULE THE BENDRIX SITUATION • Find P(A) using the relationship P(A) = P(E1)+P(E2), if the relationship is correct • Are the events E1 and E2 exhaustive?

9. MULTIPLICATION RULEINDEPENDENT EVENTS • We say that two events are independent if occurrence of one does not change the likeliness of occurrence of the other • If A and B are two independent events, the joint probabilityP(A and B) is obtained by the multiplication rule. P(A and B) = P(A)P(B)

10. CONDITIONAL PROBABILITY • Probabilities are always assessed relative to the information currently available. As new information becomes available, probabilities often change. • A formal way to revise probabilities on the basis of new information is to use conditional probabilities. • Let A and B be any events with probabilities P(A) and P(B). Typically the probability P(A) is assessed without knowledge of whether B does or does not occur. However if we are told B has occurred, the probability of A might change.

11. CONDITIONAL PROBABILITY • The new probability of A is called the conditional probability of A given B. It is denoted P(A|B). • Note that there is uncertainty involving the event to the left of the vertical bar in this notation; we do not know whether it will occur or not. However, there is no uncertainty involving the event to the right of the vertical bar; we know that it has occurred. • The following formula conditional probability formula enables us to calculate P(A|B):

12. CONDITIONAL PROBABILITY • If A and B are two mutually exclusive events, at most one of them can occur. So, P(A|B) =0 P(B|A) =0 • If A and B are two independent events, occurrence of one does not change the likeliness of occurrence of the other. So, P(A|B) = P(A) P(B|A) = P(B)

13. MULTIPLICATION RULEFOR ANY TWO EVENTS • In the conditional probability rule the numerator is the probability that both A and B occur. It must be known in order to determine P(A|B). • However, in some applications P(A|B) and P(B) are known; in these cases we can multiply both side of the conditional probability formula by P(B) to obtain the multiplication rule. P(A and B) = P(A|B)P(B) • The conditional probability formula and the multiplication rule are both valid; in fact, they are equivalent.

14. MULTIPLICATION RULE THE BENDRIX SITUATION • Are the events A and B independent? • Find P(A and B) using the multiplication rule • Does the multiplication rule give the same result as it is given by the frequencies?

15. PROBABILITY TREES • Probability trees are useful to • calculate probabilities • identify all simple events • visualize the relationship among the events • Probability trees are useful if it is possible to • break down simple events into stages • identify mutually exclusive and exhaustive events at each stage • ascertain the probabilities of events at each stage

16. PROBABILITY TREES • A probability tree consists of some nodes and branches • Nodes • an initial unlabelled node called origin • other nodes, each labeled with the event represented by the node

17. PROBABILITY TREES • Branches • each branch connect a pair of nodes. • a branch from A to B implies that event B may occur after event A • each branch from • origin to A is labeled with probability P(A) • A to B is labeled with the probability P(B|A)

18. PROBABILITY TREES • Any path through the tree from the origin to a terminal node corresponds to one possible simple event. • All simple events and their probabilities are shown next to the terminal nodes.

19. PROBABILITY TREES • Example 1: Construct a probability tree diagram for the Bendrix Company.

20. PROBABILITY TREES • Example 2: In a bag containing 7 red chips and 5 blue chips you select 2 chips one after the other without replacement. Construct a probability tree diagram for this information.

21. PROBABILITY TREES • What is the probability of getting a red chip first and then a blue chip? • What is the probability of getting a blue chip first and then a red chip? • What is the probability of getting a red and a blue chip? • What is the probability of getting 2 red chips?