Business and Finance College Principles of Statistics Eng. Heba Hamad

Slides Prepared by JOHN S. LOUCKS St. Edward’s University Slides Prepared by JOHN S. LOUCKS St. Edward’s University

Chapter 4 Introduction to Probability Experiments, Counting Rules, and Assigning Probabilities Events and Their Probability Some Basic Relationships of Probability Conditional Probability

Chapter 4 Introduction to Probability Uncertainty – everything in life Probability – a numerical measure of likelihood that an event will occur.

Examples of Uncertainty Business decisions are often based on an analysis of uncertainties such as the following: What will happen to sales if we increase prices? Will productivity increase if we change methods? How likely is it that a project will be finished on schedule?

Probability as a Numerical Measure of the Likelihood of Occurrence Increasing Likelihood of Occurrence Probability: The event is very unlikely to occur. The occurrence of the event is just as likely as just as likely as it is unlikely. The event is almost certain to occur.

Will it rain tomorrow ? A near-zero probability to rain 0.9 probability to rain 0.5 probability to rain Probability as a Numerical Measure of the Likelihood of Occurrence

Space An Experiment and Its Sample Space An experiment is any process that generates well-defined outcomes. The sample space for an experiment is the set of all experimental outcomes. An experimental outcome is also called a sample point.

Experiment and Experiment Outcomes Experiment Experiment Outcomes Toss a coin Select a part for inspection Roll a die Play a football game Head, Tail Defective, non defective 1,2,3,4,5,6 Win, Lose, Tie

Example of Experiment Toss a coin The sample space (S) = {head, tail} Sample points are 2: head, tail

Roll a die S = {1, 2, 3, 4, 5, 6} Sample space has 6 elements or sample points Example of Experiment

Toss a coin twice in succession S = {(H,H), (H,T), (T,H), (T,T)} Tree Diagram is graphical representation of the sample points in the sample space Example of Experiment

.5.5 Head Tail Head Tail Head Tail P(2 heads) =.25 P(1 head/1 tail) =.25 P(1 tail/1 head) =.25 P(2 tails) =.25

Counting rules, Combinations, and Permutations. Being able to identify and count the experimental outcomes is a necessary step in assigning probabilities. We now discuss three counting rules that are useful.

A Counting Rule for Multiple-Step Experiments If an experiment consists of a sequence of k steps in which there are n 1 possible results for the first step, n 2 possible results for the second step, and so on, then the total number of experimental outcomes is given by ( n 1 )( n 2 )... ( n k ). A helpful graphical representation of a multiple-step experiment is a tree diagram.

Example: Bradley Investments Bradley has invested in two stocks, Markley Oil and Collins Mining. Bradley has determined that the possible outcomes of these investments three months from now are as follows. Investment Gain or Loss Investment Gain or Loss in 3 Months (in $1000) in 3 Months (in $1000) Markley Oil Collins Mining  20 8 2222

Bradley Investments can be viewed as a two-step experiment. It involves two stocks, each with a set of experimental outcomes. Markley Oil: n 1 = 4 Collins Mining: n 2 = 2 Total Number of Experimental Outcomes: n 1 n 2 = (4)(2) = 8 A Counting Rule for Multiple-Step Experiments

Tree Diagram Gain 5 Gain 8 Gain 10 Gain 8 Lose 20 Lose 2 Even Markley Oil (Stage 1) Collins Mining (Stage 2) ExperimentalOutcomes (10, 8) Gain $18,000 (10, -2) Gain $8,000 (5, 8) Gain $13,000 (5, -2) Gain $3,000 (0, 8) Gain $8,000 (0, -2) Lose $2,000 (-20, 8) Lose $12,000 (-20, -2) Lose $22,000

K- Power & Light Company (KP&L) Using the counting rule in the analysis of a capacity expansion project faced by (KP&L) company. Two steps: design and construction Management cannot predict beforehand the exact time required to complete each stage of the project.

K- Power & Light Company (KP&L) An analysis of similar projects has shown the completion times for the designs stage of 2,3or4 months and completion times for construction stages of 6, 7, or 8 months. Management has set a goal of 10 months for completion of the entire project.

Since there are 3 possible completion times for design step and 3 possible completion times for construction step, the counting rule for multiple-step experiments assign the outcomes to be 3*3=9 experimental outcomes. K- Power & Light Company (KP&L)

Stage1 Stage2 Notation for E.O Total Project Comp. times K- Power & Light Company (KP&L) (2,6) (2,7) (2,8) (3,6) (3,7) (3,8) (4,6) (4,7) (4,8)

K- Power & Light Company (KP&L) 2 mo 3mo 4mo Stage 1Stage 2

The counting Rule and tree diagram have been used to help the project manager identify the experimental outcomes and determine the possible project completion times The project will be completed in from 8 to 12 months K- Power & Light Company (KP&L)