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PROBABILITY RULES AND TREES • • • • Rule of complement Addition rule Multiplication rule Probability tree 1 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 2 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? Event B BC Venn Diagram Sample Space 3 ADDITION RULE MUTUALLY 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) 4 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 5 ADDITION RULE THE BENDRIX SITUATION • Interpret the events E1 = (A and B) E2 = (A and BC) Sample Space A B Venn Diagram 6 ADDITION RULE THE BENDRIX SITUATION • Are the events E1 and E2 mutually exclusive? • Verify the following P(A) = P(E1)+P(E2) Sample Space A B Venn Diagram 7 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? Sample Space A B Venn Diagram 8 MULTIPLICATION RULE INDEPENDENT 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 probability P(A and B) is obtained by the multiplication rule. P(A and B) = P(A)P(B) 9 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. 10 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): P( A | B) P( A and B) P( B) 11 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) 12 MULTIPLICATION RULE FOR 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. 13 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 A frequencies? Sample Space B Venn Diagram 14 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 15 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 16 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) 17 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. 18 PROBABILITY TREES • Example 1: Construct a probability tree diagram for the Bendrix Company. Stage 1 P( B) =2 /3 B Stage 2 =3/4 P(A|B) P(A C|B )=1/4 C B P( C )=1/5 A BA P(BA)=0.5000 AC BAC P(BAC)=0.1667 A BCA P(BCA)=0.0667 AC BCAC P(BCAC)=0.0266 P(A|B 3 1/ )= BC Simple Joint Events Probabilities P(A C|B C )=4/5 19 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. Stage 1 Stage 2 =6/11 ) R | R ( P P( R )= 7/ 12 R P(B|R) =5 / =5 B) P( =7/11 P(R|B) /11 B Simple Joint Events Probabilities RR P(RR)=7/22 B RB P(RB)=35/132 R BR P(BR)=35/132 B BB P(BB)=5/33 12 R P(B|B) =4 /11 20 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? 21