Text of Introduction to Probability Experiments Counting Rules Combinations Permutations Assigning...
Introduction to Probability Experiments Counting Rules Combinations Permutations Assigning Probabilities
Experiments These are processes that generate well- defined outcomes ExperimentExperimental Outcomes Toss a coinHead, tail Select a part for inspection Defective, nondefective Conduct a sales callPurchase, no purchase Roll a die1, 2, 3, 4, 5, 6 Play a football gameWin, lose, tie
Probability is a numerical measure of the likelihood of an event occurring Probability: 01.0 0.5 The occurrence of the event is just as likely as it is unlikely
Sample Space The sample space for an experiment is the set of all experimental outcomes For a coin toss: Selecting a part for inspection: Rolling a die:
Counting Experimental Outcomes To assign probabilities, we must first count experimental outcomes. We have 3 useful counting rules for multiple-step experiments. For example, what is the number of possible outcomes if we roll the die 4 times? 1.Counting rule for multi-step experiments 2.Counting rule for combinations 3.Counting rule for permutations
Counting Rule for Multi-Step Experiments If an experiment can be described as a sequence of k steps with n 1 possible outcomes on the fist step, n 2 possible outcomes on the second step, then the total number of experimental outcomes is given by:
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 $000) in 3 Months (in $000) Markley Oil Collins Mining 10 5 0 20 8 2222
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
Counting Rule for Combinations This rule allows us to count the number of experimental outcomes when we select n objects from a (usually larger) set of N objects. The number of N objects taken n at a time is where And by definition
Example: Quality Control An inspector randomly selects 2 of 5 parts for inspection. In a group of 5 parts, how many combinations of 2 parts can be selected? Let the parts de designated A, B, C, D, E. Thus we could select: AB AC AD AE BC BD BE CD CE and DE
Ohio Lottery Ohio randomly selects 6 integers from a group of 47 to determine the weekly winner. What are your odds of winning if your purchased one ticket?
Counting Rule for Permutations Sometimes the order of selection matters. This rule allows us to count the number of experimental outcomes when n objects are to be selected from a set of N objects and the order of selection matters.
Example: Quality Control Again An inspector randomly selects 2 of 5 parts for inspection. In a group of 5 parts, how many permutations of 2 parts can be selected? Again let the parts de designated A, B, C, D, E. Thus we could select: AB BA AC CA AD DA AE EA BC CB BD DB BE EB CD DC CE EC DE and ED
Basic Requirements for Assigning Probabilities Let E i denote the ith experimental outcome and P(E i ) is its probability of occurring. Then: The sum of the probabilities for all experimental outcomes must be must equal 1. For n experimental outcomes:
Classical Method This method of assigning probabilities is indicated if each experimental outcome is equally likely
Example: Tossing a Die Experimental OutcomeP(E i ) 11/6 =.1667 2 3 4 5 6 P(E i )1.00
Relative Frequency Method This method is indicated when the data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times. What if experimental outcomes are NOT equally likely. Then the Classical method is out. We must assign probabilities on the basis of experimentation or historical data.
Example: Lucas Tool Rental Relative Frequency Method Lucas Tool Rental would like to assign probabilities to the number of car polishers it rents each day. Office records show the following frequencies of daily rentals for the last 40 days. Number of Polishers Rented Number of Days 0 1 2 3 4 4 6 18 10 2
Each probability assignment is given by dividing the frequency (number of days) by the total frequency (total number of days). Relative Frequency Method 4/404/40 Probability Number of Polishers Rented Number of Days 0 1 2 3 4 4 6 18 10 2 188.8.131.52.25.05 1.00
Subjective Method When economic conditions and a companys When economic conditions and a companys circumstances change rapidly it might be circumstances change rapidly it might be inappropriate to assign probabilities based solely on inappropriate to assign probabilities based solely on historical data. historical data. We can use any data available as well as our We can use any data available as well as our experience and intuition, but ultimately a probability experience and intuition, but ultimately a probability value should express our degree of belief that the value should express our degree of belief that the experimental outcome will occur. experimental outcome will occur. The best probability estimates often are obtained by The best probability estimates often are obtained by combining the estimates from the classical or relative combining the estimates from the classical or relative frequency approach with the subjective estimate. frequency approach with the subjective estimate.
Subjective Method Applying the subjective method, an analyst made the following probability assignments. Exper. Outcome Net Gain or Loss Probability (10, 8) (10, 2) (5, 8) (5, 2) (0, 8) (0, 2) ( 20, 8) ( 20, 2) $18,000 Gain $18,000 Gain $8,000 Gain $8,000 Gain $13,000 Gain $13,000 Gain $3,000 Gain $3,000 Gain $8,000 Gain $8,000 Gain $2,000 Loss $2,000 Loss $12,000 Loss $12,000 Loss $22,000 Loss $22,000 Loss.20.08.16.26.10.12.02.06