Lecture 2 Probability By Aziza Munir

Summary of last lecture Why QBA What is a model? Why to develop a model Types of models Flow chart of transformation

Learning Objectives Experiments and sample space Assigning probabilities to experimental outcomes – Classical method – Relative Frequency Method – Subjective Method Event and their Probabilities Some basic relations of Probability – Complement of Event Addition Law – Conditional Probability – Multiplication Law Summary

Probability Need: Business decisions are often based on analysis of uncertainties such as following: 1.What are the “chances” that sales will decrease if we increase profit? 2.What is the likelihood that a new assembly method will increase productivity? 3.How “likely” it is that project will be completed in time? 4.What are the “odds” in favor of the new investment being profitable?

Probability Definition: Probability is a numerical measure of the likelihood that an event will occur. Thus probabilities could be used as measures of the degree of uncertainty, that an event will occur. Probability provide a way to a.Measure b.Express c.Analyze the uncertainty associated with future events

Probability Probability values are always assigned on a scale from 0 to 1. A probability near 0 states that an event is unlikely to occur, and near 1 indicates that event is likely to occur. Other probabiliites in between describe varying degrees of likelihood.

Experiments and Sample Space Definition: An experiment is a process that generates well defined outcomes i.e. one and only one experimental outcome will occur. Consider the example…. S. NoExperimentOutcome 1Toss a coinHead/tail 2Roll a dice1,2,3,4,5,6 3Play a cricket gameWin/loose/tie 4Impact of quality control outputGood/defective 5Attend a testPass/fail 6Apply for jobAccept/decline/conditional

Probabilities of experiments We have the following laws a: 0≤P(E) ≤1 b. ∑P(E) = 1 Or P(E1)+ P(E2) +P(E3)…….P(En)=1 Of course sum of all probabilities will be non negative

Probabilities of experiments Sample Space: It indicates the maximum area where probability could occur. Consider coin tossing, possible outcomes will either be head or tail, similarly in rolling a dice it will either of 1, 2, 3, 4, 5 or 6. therefore sample space is the domain of probability occurrence. S={Head, Tail} S={1,2,3,4,5,6}

Probabilities of experiments Moreover, tossing two coins possible outcomes might be four, presented in sample space as: S={Head, tail, Head, Tail} Rolling two dice, will generate 6X6 outcomes S={(1,1)(1,2)(1,3)(1,4)(1,5)(1,6) (2,1)(2,2)(2,3)(2,4)(2,5)(2,6) (3,1)(3,2)(3,3)(3,4)(3,5)(3,6) (4,1)(4,2)($,3)(4,4)(4,5)(4,6) (5,1)(5,2)(5,3)(5,4)(5,5)(5,6) (6,1)(6,2)(6,3)(6,4)(6,5)(6,6)} 36 outcomes

Classical Method The classical method was developed originally to analyse gambling probabilities, where assumption of equally likely outcomes often is reasonable. Consider similar example of tossing a coin, where chance of getting head or appearing tail is equally likely, as the outcomes may either be head or tail with equal chance of appearance then we can say that probability to get head as outcome is 0.50 or ½ and similar is with tail appearance. P(H)=1/2 P(T)=1/2

Relative Frequency method Classical method has multiple limitations, towards scope, therefore alternative means have been developed Relative frequency method describes the ratio of successive chances to occur and total number of outcome. P (E) = S/T Example: 100 consumers buy a product from total production of 400 P (E)= 100/400=0.25

Subjective Method The classical and relative frequency methods of assigning probabilities are objective. For the same experiment or data we should agree on the probability assignments. Subjective method, involves the personal degree of belief. Different individuals looking at same experiment can provide equally good but different subjective probabilities. e.g. in a game, winning, losing or tie wont have equal chance of occurance.

Some basic relations of probability Complement of an event S-A=A` P(A)+ P(A`)=1 A

Union and intersection AUB AПB S B

Addition Law P(AUB)=P(A)+P(B)-P(AnB) Example: of 200 students taking a course, 160 passed mid term exam, 140 passed final exam and 124 passed both. A= event of passing mid term exam B= event of passing final exam P(A)= 160/200=0.80 P(B)=140/200=0.70 P(AnB)=124/200=0.62 P(AUB)= =0.88

Summary Probability Experiment and sample space Assigning probabilities Simple theory of sets Addition law of probability