LESSON 5: PROBABILITY Outline Probability Events Complementary events, rule of complement Mutually exclusive and exhaustive events Addition law Independent events Conditional probability Multiplication law

PROBABILITY Concept of probability is quite intuitive However, the rules of probability are not always intuitive or easy to master. Mathematically, a probability is a number between 0 and 1 that measures the likelihood that some event will occur. An event with probability zero cannot occur. An event with probability 1 is certain to occur. An event with probability greater than 0 and less than 1 involves uncertainty, but the closer its probability is to 1 the more likely it is to occur.

BENDRIX COMPANY’S SITUATION The Bendrix company supplies contractors with materials for the construction of houses. Bendrix currently has a contract with one of its customers to fill an order by the end of July. There is uncertainty about whether this deadline can be met, due to uncertainty about whether Bendrix will receive the materials it needs from one of its suppliers by the middle of July. It is currently July 1. How can the uncertainty in this situation be assessed?

BENDRIX COMPANY’S SITUATION Bendrix collects their records on the same supplier and similar contracts and the data is shown on right:

EVENTS Events: A preliminary step is to catalog the events that may occur. Example: A: Bendrix meets due date AC: Bendrix does not meet due date B: The supplier meets due date BC: The supplier does not meet due date

COMPLEMENTARY EVENTS Complement of an event A is the event that A does not occur. Complement of an event A is denoted by AC or or not A.

PROBABILITY: THE BENDRIX SITUATION Compute the likelihood that Bendrix will meet the due date of the contract. P(A)

PROBABILITY: THE BENDRIX SITUATION Compute the likelihood that the supplier will meet the due date. P(B)

PROBABILITY: THE BENDRIX SITUATION Compute the likelihood that the due date of the contract will be made if the supplier meets the due date. Probability(A occurs given that B has occurred) P(A|B)

PROBABILITY: THE BENDRIX SITUATION Compute the likelihood that the due date of the contract will be made even if the supplier fails to meet the due date. Probability(A occurs given that BC has occurred) P(A|BC)

PROBABILITY: THE BENDRIX SITUATION Compute the likelihood that both the contract due date and supplier due date will be met. Probability(A and B both occur) P(A and B)

PROBABILITY: THE BENDRIX SITUATION Compute the likelihood that either the contract due date or the supplier due date will be met. Probability(A occurs or B occurs or both occur) P(A or B)

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

RULE OF COMPLEMENT THE BENDRIX SITUATION Summarize the following probabilities: P(B) P(Bc)

RULE OF COMPLEMENT THE BENDRIX SITUATION To see the validity of the rule of complements, check if P(Bc)=1- P(B)

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) P(A1 or A2 or A3 or A4 … or An) = P(A1) + P(A2) +  + P(An)

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 mutually exclusive and exhaustive, P(A)+ P(B)=1 If A, B and C are mutually exclusive and exhaustive, P(A)+ P(B)+ P(C)=1

ADDITION RULE THE BENDRIX SITUATION Interpret the events E1 = (A and B) E2 = (A and BC) Check if (E1 or E2) ≡ A

ADDITION RULE THE BENDRIX SITUATION Summarize the following probabilities: P(A) P(E1) = P(A and B) P(E2) = P (A and BC)

ADDITION RULE THE BENDRIX SITUATION Are the events E1 and E2 mutually exclusive? Are the events E1 and E2 exhaustive? To see the validity of the addition law for mutually exclusive events, check if P(E1 or E2) = P(E1)+P(E2) P(A) = P(E1)+P(E2)

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)

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.

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.

CONDITIONAL PROBABILITY The following conditional probability formula enables us to calculate P(A|B):

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)

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.

MULTIPLICATION RULE THE BENDRIX SITUATION Summarize the following probabilities: P(A) P(B) P(A and B) P(A|B)

MULTIPLICATION RULE THE BENDRIX SITUATION Are the events A and B independent? Which formula applies? Check with numerical values. a. P(A and B) = P(A)P(B) b. P(A and B) = P(A|B)P(B)

READING AND EXERCISES Lesson 5 Reading: Sections 6-1, 6-2, 6-3, pp. 156-171 Exercises: 6-4, 6-12, 6-13, 6-17, 6-23