Chapter 4 Probability

Probability Defined A probability is a number between 0 and 1 that measures the chance or likelihood that some event or set of events will occur.

Assigning Basic Probabilities Classical Approach Relative Frequency Approach Subjective Approach

Classical Approach P(A) = where P(A) = probability of event A F = number of outcomes “favorable” to event A T = total number of outcomes possible in the experiment

Relative Frequency Approach P(A) = where N = total number of observations or trials n = number of times that event A occurs

The Language of Probability Simple Probability Conditional Probability Independent Events Joint Probability Mutually Exclusive Events Either/Or Probability

Statistical Independence Two events are said to be statistically independent if the occurrence of one event has no influence on the likelihood of occurrence of the other.

Statistical Independence (4.1) P(A l B) = P(A) and P (B l A) = P(B) “given”

General Multiplication Rule (4.2) P(A  B) = P(A)∙ P(BlA) “and”

Multiplication Rule for (4.3) Independent Events P(A  B) = P(A) ∙P(B)

Mutually Exclusive Events Two events, A and B, are said to be mutually exclusive if the occurrence of one event means that the other event cannot or will not occur.

Mutually Exclusive Events (4.4) P(A  B) = 0

General Addition Rule (4.5) P(A  B) = P(A) + P(B) - P(A  B) “or”

Addition Rule for (4.6) Mutually Exclusive Events P(A  B) = P(A) + P(B)

“Conditional Equals JOINT Over SIMPLE” Rule P(B I A) = (4.7a) P(A I B) = (4.7b)

Complementary Events Rule (4.8) P(A′ ) = 1 – P(A)

Figure 4.1 Venn Diagram for the Internet Shoppers Example A(.8) (Airline Ticket Purchase) B(.6) (Book Purchase) A∩ B (.5) The sample space contains 100% of the possible outcomes in the experiment. 80% of these outcomes are in Circle A; 60% are in Circle B; 50% are in both circles. Sample Space (1.0)

Figure 4.2 Complementary Events The events A and A’ are said to be complementary since one or the other (but never both) must occur. For such events, P(A’ ) = 1 - P(A). Sample Space (1.0) A.8 A’ (.2) (everything in the sample space outside A)

Figure 4.3 Mutually Exclusive Events Mutually exclusive events appear as non-overlapping circles in a Venn diagram. Sample Space (1.0) BA

Figure 4.4 Probability Tree for the Project Example B is not under budget A A' A is under budget B is under budget B B' A is not under budget B is under budget B is not under budget B PROJECT A PERFORMANCE PROJECT B PERFORMANCE B' STAGE 1STAGE 2

Figure 4.5 Showing Probabilities on the Tree B is not under budget A (.25) A‘(.75) A is under budget B is under budget B(.6) B‘(.4) A is not under budget B is under budget B is not under budget B(.2) PROJECT A PERFORMANCE PROJECT B PERFORMANCE B‘(.8) STAGE 1STAGE 2

Figure 4.6 Identifying the Relevant End Nodes On The Tree  B is not under budget A (.25) A‘(.75) A is under budget B is under budget B(.6) B‘(.4) A is not under budget B is under budget B is not under budget B(.2) PROJECT A PERFORMANCE PROJECT B PERFORMANCE B‘(.8) STAGE 1STAGE 2 (2) (1) (3) (4) 

Figure 4.7 Calculating End Node Probabilities B is not under budget A (.25) A‘(.75) A is under budget B is under budget B(.6) B‘(.4) A is not under budget B is under budget B is not under budget B(.2) PROJECT A PERFORMANCE PROJECT B PERFORMANCE B‘(.8) STAGE 1STAGE 2 (2) (1) (3) (4).10.15

Figure 4.8 Probability Tree for the Spare Parts Example Unit is OK A 1 A 2 Adams is the supplier Unit is defective B B' Alder is the supplier Unit is defective Unit is OK B B' (.7) (.3) (.04) (.96) (.07) (.93) SOURCECONDITION

Figure 4.9 Using the Tree to Calculate End-Node Probabilities Unit is OK A 1 A 2 Adams is the supplier Unit is defective B B' Alder is the supplier Unit is defective Unit is OK B B' (.7) (.3) (.04 ) (.96) (.07) (.93) SOURCECONDITION .028 .021

Bayes’ Theorem (4.9) (Two Events) P(A 1 l B) =

General Form of Bayes’ Theorem P ( A i l B ) =

Cross-tabs Table Very ImportantImportantNot Important Under Grad Grad

Joint Probability Table Very Important Not Important Under Grad Grad

Counting Total Outcomes (4.10) in a Multi-Stage Experiment Total Outcomes = m 1 x m 2 x m 3 x…x m k where m i = number of outcomes possible in each stage k = number of stages

Combinations (4.11) n C x = where n C x = number of combinations (subgroups) of n objects selected x at a time n = size of the larger group x = size of the smaller subgroups

Figure 4.10 Your Car and Your Friends Five friends are waiting for a ride in your car, but only four seats are available. How many different arrangements of friends in the car are possible? A B E C D 1 Friends Car

Permutations (4.12) nPx =nPx =