Business Statistics A Decision-Making Approach 8 th Edition

Business Statistics: A Decision-Making Approach 8 th Edition Chapter 5 Discrete Probability Distributions 5 -1

Chapter Goals After completing this chapter, you should be able to: n n Calculate and interpret the expected value of a discrete probability distribution Apply the binomial distribution to business problems Compute probabilities for the Poisson and hypergeometric distributions Recognize when to apply discrete probability distributions to decision making situations 5 -2

Probability Distributions Ch. 5 Ch. 6 Discrete Probability Distributions Continuous Probability Distributions Binomial Normal Poisson Uniform Hypergeometric Exponential 5 -3

Random variable vs. Probability distribution n n When the value of a variable is the outcome of a statistical experiment, that variable is a random variable. A probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence. 5 -4

Random variable vs. Probability distribution Experiment: Toss 2 Coins. Let x = # heads. 4 possible outcomes T T x Value Probability 0 1/4 = 0. 25 T H 1 2/4 = 0. 50 2 1/4 = 0. 25 H T H H 5 -5

Cumulative Probability and Cumulative Probability Distribution n A cumulative probability refers to the probability that the value of a random variable falls within a specified range. n n Probability for at most (less than equal to: <) one head? 0. 25+0. 5=0. 75 A cumulative probability distribution can be represented by a table or an equation. 5 -6

Cumulative Probability Distribution Cumulative Number of heads: x Probability: P(X = x) Probability: P(X < x) 0 0. 25 1 0. 50 0. 75 2 0. 25 1. 00 5 -7

Discrete Probability Distribution n If a random variable is a discrete variable, its probability distribution is called a discrete probability distribution. Number of heads Probability 0 0. 25 1 0. 50 2 0. 25 5 -8

Mean (formula) n Expected Value (or mean) of a discrete probability distribution (Weighted Average – e. g. , GPA) E(x) = n x. P(x) Example: Toss 2 coins, x = # of heads, compute expected value of x: E(x) = (0 x 0. 25) + (1 x 0. 50) + (2 x 0. 25) x P(x) 0. 25 1 0. 50 2 0. 25 = 1. 0 5 -9

Standard Deviation (formula) n Standard Deviation of a discrete probability distribution where: E(x) = Expected value of the random variable (done!) x = Values of the random variable P(x) = Probability of the random variable having the value of x 5 -10

Standard Deviation (continued) n Example: Toss 2 coins, x = # heads, compute standard deviation (recall E(x) = 1) Possible number of heads = 0, 1, or 2 5 -11

Using Excel n n Review real world business examples on page 194 and page 195 Use Excel for calculating: n n n Discrete Random Variable Mean Discrete Random Variable Standard Deviation Download and open “Binomial Poisson Distribution” Excel file… n And then, try the example on the first tap… 5 -12

Binomial Experiment n n n The experiment involves repeated trials. Each trial has only two possible outcomes - a success or a failure (i. e. , head/tail, goal/no goal). The probability that a particular outcome will occur on any given trial is constant. n n 0. 5 every trial All of the trials in the experiment are independent. n The outcome on one trial does not affect the outcome on other trials. 5 -13

Binomial Experiment Example Outcome, x Binomial probability, P(X = x) Cumulative probability, P(X < x) 0 Heads 0. 125 1 Head 0. 375 0. 500 2 Heads 0. 375 0. 875 3 Heads 0. 125 1. 000 5 -14

Binomial Probability n n A binomial probability refers to the probability of getting EXACTLY n successes in a specific number of trials. Example: What is the probability of getting EXACTLY 2 Heads in 3 coin tosses. n Using the table on the previous slide, that probability (0. 375) would be an example of a binomial probability. 5 -15

Cumulative Binomial Probability n n Cumulative binomial probability refers to the probability that the value of a binomial random variable falls within a specified range. Example: What is the probability of getting AT MOST 2 Heads (meaning, less than equal to: <) in 3 coin tosses is an example of a cumulative probability. n n 0 heads (0. 125) + 1 head (0. 375) + 2 heads (0. 375). Thus, the cumulative probability of getting AT MOST 2 Heads in 3 coin tosses is equal to 0. 875. 5 -16

Translation to Math Notations n n The probability of getting FEWER (LESS) THAN 2 successes is indicated by P(X < 2). The probability of getting AT MOST 2 successes is indicated by P(X < 2). The probability of getting AT LEAST 2 successes is indicated by P(X > 2). The probability of getting MORE (GREATER) THAN 2 successes is indicated by P(X > 2). 5 -17

Binomial Distribution n The shape of the binomial distribution depends on the values of p and n Mean n Here, n = 5 and p = 0. 1 Try the “Binomial Distribution Simulation” on the class website n Here, n = 5 and p = 0. 5 . 6. 4. 2 0 P(X) X 0 . 6. 4. 2 0 n = 5 p = 0. 1 P(X) 1 2 3 4 5 n = 5 p = 0. 5 X 0 1 2 3 4 5 5 -18

Binomial Distribution Formula n! x n-x P(x) = p q x ! (n - x ) ! P(x) = probability of x successes in n trials, with probability of success p on each trial x = number of successes in sample, (x = 0, 1, 2, . . . , n) p = probability of “success” per trial q = probability of “failure” = (1 – p) n = number of trials (sample size) 5 -19

Binomial Distribution Example: 35% of all voters support Proposition A. If a random sample of 10 voters is polled, what is the probability that exactly three of them support the proposition? i. e. , find P(x = 3) if n = 10 and p = 0. 35 : There is a 25. 22% chance that 3 out of the 10 voters will support Proposition A 5 -20

Using Excel n Try the binominal distribution using Excel… n n n Download and open “Binomial Poisson Distribution” Excel file… Try followings together; n Binom-1 n Binom-2 n Binom-3 Then, try exercise 5 -34 and 5 -36 with your neighboor 5 -21

The Poisson Distribution n The Poisson Distribution is a discrete distribution which takes on the values X = 0, 1, 2, 3, . . It is often used as a model for the number of events in a specific time period. Events examples: n n the number of telephone calls at a call center the number of bags lost per flight 5 -22

Poisson Distribution Summary Measures n n Mean Variance and Standard Deviation where = number of successes in a segment of unit size t = the size of the segment of interest 5 -23

Poisson Distribution Formula where: t = size of the segment of interest x = number of successes in segment of interest = expected number of successes in a segment of unit size e = base of the natural logarithm system (2. 71828. . . ) Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 5 -24

Example n n The average number of homes sold by the Acme Realty company is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow? Solution: This is a Poisson experiment in which we know the following: n n n μ = 2; since 2 homes are sold per day, on average. x = 3; since we want to find the likelihood that 3 homes will be sold tomorrow. e = 2. 71828; since e is a constant equal to approximately 2. 71828. 5 -25

Example (con’t) n We plug these values into the Poisson formula as follows: n n P(3; 2) = (23) (2. 71828 -2) / 3! P(3; 2) = (0. 13534) (8) / 6 P(3; 2) = 0. 180 Thus, the probability of selling 3 homes tomorrow is 0. 180. 5 -26

Poisson distribution – Using Excel n n n Excel can be used to find both the cumulative probability as well as the point estimated probability for a Poisson experiment. In order to get Excel to calculate poisson probabilities, you have to use the following syntax in a cell. =poisson (x; mean; cumulative) Previous example n =poisson (2; 3; false) = 0. 180 5 -27

Poisson distribution – Using Excel n n n X is the number of events. Mean is simply the mean of the variable. Cumulative has the options of FALSE and TRUE. n n If you choose FALSE, Excel will return probability of only and only the x number of events happening. If you choose TRUE, Excel will return the cumulative probability of the event x or less happening. 5 -28

Example: Point Estimate n n n A bakery has average 6 customers during a business hour. The bakery wishes to calculate the probability of the event that exactly 4 customers enter the store in the next hour. That is: x = 4, mean = 6 and cumulative = FALSE Would be written in excel as: =poisson(4; 6; FALSE) And return the probability of 0. 133853 = 13. 3853% 5 -29

Example: Cumulative n n n A bakery has average 6 customers during a business hour. We then wish to calculate the probability of the event that 4 customers or less enter the store in the next hour. That is: x = 4, mean = 6 and cumulative = TRUE Would be written in excel as: =poisson(4; 6; TRUE) And return the probability of 0. 285057 = 28. 5057% 5 -30

Using Excel n Try the poisson distribution using Excel… n n Download and open “Binomial Poisson Distribution” Excel file… Try “Poisson”……… 5 -31

Poisson Distribution Heritage Title Try this……. Issue: The distribution for the number of defects per tile made by Heritage Tile is Poisson distributed with a mean of 3 defects per tile. The manager is worried about the high variability Objective: Use Excel 2007 or 2010 to generate the Poisson distribution and histogram to visually see spread in the distribution of possible defects. 5 -32

Poisson Distribution – Heritage Tile Enter values zero through 10 5 -33

Poisson Distribution – Heritage Tile Select Formulas, More Functions, Statistical and POISSON 5 -34

Poisson Distribution – Heritage Tile Enter: a 1, 3, false 5 -35

Poisson Distribution – Heritage Tile Notice that I had pre-selected Cell B 1. When I pressed enter the Poisson Probability was loaded into that cell. Simply copy and paste Cell B 1 into cells B 2 : B 11 5 -36

Poisson Distribution – Heritage Tile • Select the Insert tab • Select Column • Select the chart type that you want 5 -37

Poisson Distribution – Heritage Tile Format the chart as per Chapter 2 5 -38

Chapter Summary n Reviewed key discrete distributions n n n Binomial Poisson Hypergeometric n Found probabilities using formulas and tables n Recognized when to apply different distributions n Applied distributions to decision problems 5 -39