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CHAPTER 6: PROBABILITY PART I Outline • • • • • • Introduction Probability rules and trees Probability distributions Expected value and variance Binomial distribution Poisson distribution 1 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? 2 SOME KEY TERMS (DEFINITION) • Random experiment, events, simple events, sample space, and complement: – A random experiment is a process of generating (simple) events. The (simple) events generated by a random experiment cannot be predicted with certainty – Simple events cannot be broken down, or decomposed, into two or more constituent outcomes – An event is a collection of the simple events of interest. So, a simple event is also an event. – The sample space is the list of all possible events 3 SOME KEY TERMS (DEFINITION) – Complement of an event A is the event that A does not occur. Sometimes, complement of an event is not just an event, but a set of many events. So, a more precise definition is that the complement of an event A is the set of all events other than A. Complement of an event A is denoted by AC or A Event A AC Venn Diagram Sample Space 4 SOME KEY TERMS (EXAMPLE) – Consider the problem of finding a simple random sample of 10 families from a list 40 families given in file RANDSAMP.XLS – The process of finding a simple random sample is a random experiment • The process generated a simple random sample. • A (simple) event is a sample consisting of families 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. Another (simple) event is a sample consisting of families 1, 2, 3, 4, 5, 6, 7, 8, 9 and 11. 5 SOME KEY TERMS (EXAMPLE) • There is a large number of ways (how many?) we can choose 10 families from a list of 40 families. The sample space consists of all possible combination of 10 families e.g., 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 1, 2, 3, 4, 5, 6, 7, 8, 9, 11 1, 2, 3, 4, 5, 6, 7, 8, 9, 12 … 6 SOME KEY TERMS (EXAMPLE) • Example of an event: A sample of 10 families that includes 5 families with above average income and 5 families with below average income – this event is a subset of sample space – this event is a collection of events – notice that a simple event is an even, but an event is not necessarily a simple event 7 SOME KEY TERMS (EXAMPLE) • Let (simple) event A be the sample consisting of families 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. It’s complement is the set of all possible combination of 10 families except A e.g., 1, 2, 3, 4, 5, 6, 7, 8, 9, 11 1, 2, 3, 4, 5, 6, 7, 8, 9, 12 1, 2, 3, 4, 5, 6, 7, 8, 9, 13 … 8 PROBABILITY ESSENTIALS • 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. 9 Supplier meets due date BENDRIX COMPANY’S SITUATION • Bendrix collects their records on the same supplier and similar contracts and the data is shown on right: Bendrix meets due date Event B Supplier does not meet due date 30 4 34 10 16 26 40 20 60 Total Event BC Event A Bendrix does not meet due date Event AC Total 10 PROBABILITY ESSENTIALS • Consider the Bendrix Company data. • Compute the likelihood that the due date of the contract will be met. Probability(A occurs), P(A) • Compute the likelihood that the supplier will meet the due date. Probability(B occurs), P(B) 11 PROBABILITY ESSENTIALS • Consider the Bendrix Company data. • 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) 12 PROBABILITY ESSENTIALS • Consider the Bendrix Company data. • 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) 13 PROBABILITY ESSENTIALS • Consider the Bendrix Company data. • 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) 14 PROBABILITY ESSENTIALS • Consider the Bendrix Company data. • 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) 15 PROBABILITY RULES AND TREES • • • • Rule of complement Addition rule Multiplication rule Probability tree 16 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 17 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 18 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) 19 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 20 ADDITION RULE THE BENDRIX SITUATION • Interpret the events E1 = (A and B) E2 = (A and BC) Sample Space A B Venn Diagram 21 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 22 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 23 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) 24 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. 25 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) 26 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) 27 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. 28 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 29 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 30 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 31 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) 32 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. 33 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( 3 1/ )= BC Simple Joint Events Probabilities 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 P(A C|B C )=4/5 34 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) =5/11 / =5 B) P( =7/11 P(R|B) 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 35 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? 36