Introduction to Probability Uncertainty Probability Tree Diagrams Combinations

Introduction to Probability Uncertainty, Probability, Tree Diagrams, Combinations and Permutations Chapter 4 BA 201 Slide 1

PROBABILITY Slide 2

Uncertainty Managers often base their decisions on an analysis of uncertainties such as the following: What are the chances that sales will decrease if we increase prices? What is the likelihood a new assembly method will increase productivity? What are the odds that a new investment will be profitable? Slide 3

Probability is a numerical measure of the likelihood that an event will occur. Probability values are from 0 to 1. Slide 4

Probability as a Numerical Measure of the Likelihood of Occurrence Increasing Likelihood of Occurrence Probability: 0 The event is very unlikely to occur. 0. 5 The occurrence of the event is just as likely as it is unlikely. 1 The event is almost certain to occur. Slide 5

STATISTICAL EXPERIMENTS Slide 6

Statistical Experiments In statistical experiments, probability determines outcomes. Even though the experiment is repeated in exactly the same way, an entirely different outcome may occur. Slide 7

An Experiment and Its Sample Space An experiment is any process that generates welldefined outcomes. The sample space for an experiment is the set of all experimental outcomes. An experimental outcome is also called a sample point. Roll a die 1 2 3 4 5 6 Slide 8

An Experiment and Its Sample Space Experiment Outcomes Toss a coin Head, tail Inspect a part Defective, non-defective Conduct a sales call Purchase, no purchase Slide 9

An Experiment and Its Sample Space 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 in 3 Months (in $000) Collins Mining Markley Oil 8 10 -2 5 0 -20 Slide 10

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: # outcomes = (n 1)(n 2). . . (nk ) Slide 11

A Counting Rule for Multiple-Step Experiments Bradley Investments can be viewed as a two-step experiment. It involves two stocks, each with a set of experimental outcomes. Markley Oil: Collins Mining: n 1 = 4 n 2 = 2 Total Number of Experimental Outcomes: n 1 n 2 = (4)(2) = 8 Slide 12

Tree Diagram Bradley Investments Markley Oil (Stage 1) Collins Mining (Stage 2) Gain 8 Gain 10 Gain 8 Gain 5 Lose 2 Gain 8 Even Lose 20 Gain 8 Lose 2 Experimental Outcomes (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 Slide 13

Counting Rule for Combinations Number of Combinations of N Objects Taken n at a Time Combinations enable us to count the number of experimental outcomes when n objects are to be selected from a set of N objects. where: N! = N(N - 1)(N - 2). . . (2)(1) n! = n(n - 1)(n - 2). . . (2)(1) 0! = 1 Slide 14

Counting Rule for Permutations Number of Permutations of N Objects Taken n at a Time Permutations enable us to count the number of experimental outcomes when n objects are to be selected from a set of N objects, where the order of selection is important. where: N! = N(N - 1)(N - 2). . . (2)(1) n! = n(n - 1)(n - 2). . . (2)(1) 0! = 1 Slide 15

Combinations and Permutations 4 Objects: A B C D AB BC AC BD AD CD AB BA AC CA AD DA BC CB BD DB CD DC Slide 16

PRACTICE TREE DIAGRAMS, COMBINATIONS, AND PERMUTATIONS Slide 17

Practice Tree Diagram A box contains six balls: two green, two blue, and two red. You draw two balls without looking. How many outcomes are possible? Draw a tree diagram depicting the possible outcomes. Slide 18