BCOR 1020 Business Statistics Lecture 6 – February 5, 2007

Overview Chapter 4 Example Chapter 5 – Probability –Random Experiments –Probability

Chapter 4 - Example Problem 4.22 list the rents paid by a random sample of 30 students who live off campus. The sorted data is below. Using Excel, we can quickly calculate the sample average and standard deviation… Using these, we can find standardized values (z i )… Find the Quartiles and Construct a Boxplot…

Chapter 4 - Example Ordered Data… Median = 720 Q1 = 660 Q3 = 760 IQR =

Chapter 5 - Probability

Chapter 5 – Random Experiments Sample Space: A random experiment is an observational process whose results cannot be known in advance and whose outcomes will differ based on random chance. The sample space (S) for the experiment is the set of all possible outcomes in the experiment. –A discrete sample space is one with a countable (but perhaps infinite) number of outcomes. –A continuous sample space is one where the outcomes fall on a continuous interval (often the result of a measurement).

Chapter 5 – Random Experiments Sample Space: Discrete Sample Space Examples: The sample space describing a Wal-Mart customer’s payment method is… S = {cash, debit card, credit card, check} Continuous Sample Space Examples: The sample space for the length of a randomly chosen cell phone call would be… S = {all X such that X > 0} or written as S = {X | X > 0}. The sample space to describe a randomly chosen student’s GPA would be S = {X | 0.00 < X < 4.00}.

Chapter 5 – Random Experiments Some sample spaces can be enumerated: For Example: –For a single roll of a die, the sample space is: S = {1, 2, 3, 4, 5, 6}. –When two dice are rolled, the sample space is the following pairs: {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} S =

Chapter 5 – Random Experiments Some sample spaces are not easily enumerated: For Example: –Consider the sample space to describe a randomly chosen United Airlines employee by 6 home bases (major hubs),2 genders, 21 job classifications,4 education levels –There are: 6 x 22 x 21 x 4 = 1008 possible outcomes.

Chapter 5 – Random Experiments Events: An event is any subset of outcomes in the sample space. A simple event or elementary event, is a single outcome. –A discrete sample space S consists of all the simple events (E i ): S = {E1, E2, …, En} A compound event consists of two or more simple events.

Chapter 5 – Random Experiments Example of a Simple Event: Consider the random experiment of tossing a balanced coin. What is the sample space? S = {H, T} What are the chances of observing a H or T? These two elementary events are equally likely.

Clickers When you buy a lottery ticket, the sample space S = {win, lose} has only two events. Are these two events equally likely to occur? A = Yes B = No

Chapter 5 – Random Experiments Example of Compound Events: Recall: a compound event consists of two or more simple events. For example, in a sample space of 6 simple events, we could define the compound events… These are displayed in a Venn diagram: A = {E 1, E 2 } B = {E 3, E 5, E 6 } Many different compound events could be defined. Compound events can be described by a rule.

Clickers Recall our earlier example involving the roll of two dice where the sample space is given by… If we define the compound event A = “rolling a seven” on a roll of two dice, how many simple events does our compound event consist of? A = 4B = 6 C = 7D = 36 {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} S =

Chapter 5 – Probability Definition: The probability of an event is a number that measures the relative likelihood that the event will occur. –The probability of event A [denoted P(A)], must lie within the interval from 0 to 1: 0 < P(A) < 1 If P(A) = 0, then the event cannot occur. If P(A) = 1, then the event is certain to occur.

Chapter 5 – Probability Definitions: In a discrete sample space, the probabilities of all simple events must sum to unity: P(S) = P(E1) + P(E2) + … + P(En) = 1 For example, if the following number of purchases were made by… credit card: 32% debit card: 20% cash: 35% check: 18% Sum =100% Probability P(credit card) =.32 P(debit card) =.20 P(cash) =.35 P(check) =.18 Sum =1.0

Chapter 5 – Probability Businesses want to be able to quantify the uncertainty of future events. –For example, what are the chances that next month’s revenue will exceed last year’s average? The study of probability helps us understand and quantify the uncertainty surrounding the future. –How can we increase the chance of positive future events and decrease the chance of negative future events?

Chapter 5 – Probability What is Probability? There are three approaches to probability: –Empirical– Classical– Subjective Empirical Approach: Use the empirical or relative frequency approach to assign probabilities by counting the frequency ( f i ) of observed outcomes defined on the experimental sample space. For example, to estimate the default rate on student loans P(a student defaults) = f /n number of defaults number of loans =

Chapter 5 – Probability Empirical Approach: Necessary when there is no prior knowledge of events. As the number of observations (n) increases or the number of times the experiment is performed, the estimate will become more accurate. Law of Large Numbers: The law of large numbers is an important probability theorem that states that a large sample is preferred to a small one.

Chapter 5 – Probability Example: Law of Large Numbers: Flip a coin 50 times. We would expect the proportion of heads to be near.50. –However, in a small finite sample, any ratio can be obtained (e.g., 1/3, 7/13, 10/22, 28/50, etc.). –A large n may be needed to get close to.50. Consider the results of simulating 10, 20, 50, and 500 coin flips…

Chapter 5 – Probability

Classical Approach: In this approach, we envision the entire sample space as a collection of equally likely outcomes. Instead of performing the experiment, we can use deduction to determine P(A). a priori refers to the process of assigning probabilities before the event is observed. a priori probabilities are based on logic, not experience.

Chapter 5 – Probability Classical Approach: For example, the two dice experiment has 36 equally likely simple events. The probability that the sum of the two dice is 7, P(7), is The probability is obtained a priori using the classical approach as shown in this Venn diagram for 2 dice:

Clickers Consider the Venn Diagram for the roll of two dice from the previous example: What is the probability that the two dice sum to 4, P(4)? A = B = C = D = E = 0.194

Chapter 5 – Probability Subjective Approach: A subjective probability reflects someone’s personal belief about the likelihood of an event. –Used when there is no repeatable random experiment. –For example, What is the probability that a new truck product program will show a return on investment of at least 10 percent? What is the probability that the price of GM stock will rise within the next 30 days? –These probabilities rely on personal judgment or expert opinion. Judgment is based on experience with similar events and knowledge of the underlying causal processes.