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

Probability

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 method will increase productivity? What are the odds that a new investment will be profitable?

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

Probability as a Numerical Measure of the Likelihood of Occurrence Increasing Likelihood of Occurrence 0.5 1 Probability: The event is very unlikely to occur. The occurrence of the event is just as likely as it is unlikely. The event is almost certain to occur.

Statistical Experiments

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.

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. Roll a die 1 2 3 4 5 6

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

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) Markley Oil Collins Mining 10 5 -20 8 -2

A Counting Rule for Multiple-Step Experiments If an experiment consists of a sequence of k steps in which there are n1 possible results for the first step, n2 possible results for the second step, and so on, then the total number of experimental outcomes is given by: # outcomes = (n1)(n2) . . . (nk)

A Counting Rule for Multiple-Step Experiments Bradley Investments Bradley Investments can be viewed as a two-step experiment. It involves two stocks, each with a set of experimental outcomes. Markley Oil: n1 = 4 Collins Mining: n2 = 2 Total Number of Experimental Outcomes: n1n2 = (4)(2) = 8

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

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. Order does not matter – which means a combination lock is really a misnomer. Without replacement/repetition. where: N! = N(N - 1)(N - 2) . . . (2)(1) n! = n(n - 1)(n - 2) . . . (2)(1) 0! = 1

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. Order matters – so why is it not called a permutation lock? Without replacement/repetition. where: N! = N(N - 1)(N - 2) . . . (2)(1) n! = n(n - 1)(n - 2) . . . (2)(1) 0! = 1

Combinations and Permutations 4 Objects: A B C D A B B A A C C A A B B C A D D A A C B D B C C B B D D B A D C D C D D C

Practice Tree Diagrams, Combinations, and Permutations

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. GG GB GR RG RB RR BG BB BR

Combinations There are five boxes numbered 1 through 5. You pick two boxes. How many combinations of boxes are there? Show the combinations. 1 2 1 3 1 4 1 5 2 3 2 4 2 5 3 4 3 5 4 5

Combinations There are five boxes numbered 1 through 5. You pick two boxes. How many permutations of boxes are there? Show the permutations. 1 2 1 3 1 4 1 5 2 1 3 1 4 1 5 1 2 3 2 4 2 5 3 2 4 2 5 2 3 4 3 5 4 3 5 3 4 5 5 4