Sherlock Holmes once observed that men are insoluble puzzles except in the aggregate, where they become mathematical certainties.

“You can never foretell what any one man will do,” observed Holmes, “but you can say with precision what an average number will be up to.

Individuals vary, but percentages remain constant. So says the statistician.”

Basic Probability & Discrete Probability Distributions Why study Probability?

To infer something about the population based on sample observations We use Probability Analysis to measure the chance that something will occur.

What’s the chance If I flip a coin it will come up heads?

50-50 If the probability of flipping a coin is 50-50, explain why when I flipped a coin, six of the tosses were heads and four of the tosses were tails?

Think of probability in the long run: A coin that is continually flipped, will 50% of the time be heads and 50% of the time be tails in the long run.

Probability is a proportion or fraction whose values range between 0 and 1, inclusively.

The Impossible Event Has no chance of occurring and has a probability of zero.

The Certain Event Is sure to occur and has a probability of one.

Probability Vocabulary 1)Experiment 2)Events 3)Sample Space 4)Mutually Exclusive 5)Collectively Exhaustive 6)Independent Events 7)Compliment 8)Joint Event

Experiment An activity for which the outcome is uncertain.

Examples of an Experiment: Toss a coin Select a part for inspection Conduct a sales call Roll a die Play a football game

Events Each possible outcome of the experiment.

Examples of an Event: Toss a coin Select a part for inspection Conduct a sales call Roll a die Play a football game Heads or tails Defective or non- defective Purchase or no purchase 1,2,3,4,5,or 6 Win, lose, or tie

Sample Space The set of ALL possible outcomes of an experiment.

Examples of Sample Spaces: Toss a coin Select a part for inspection Conduct a sales call Roll a die Play a football game Heads, tails Defective, nondefective Purchase, no purchase 1,2,3,4,5,6 Win, lose, tie

Mutually Exclusive Events cannot both occur simultaneously.

Collectively Exhaustive A set of events is collectively exhaustive if one of the events must occur.

Independent Events If the probability of one event occurring is unaffected by the occurrence or nonoccurrence of the other event.

Complement The complement of Event A includes all events that are not part of Event A. The complement of Event A is denoted by Ā or A’. Example: The compliment of being male is being female.

Joint Event Has two or more characteristics.

Probability Vocabulary 1)Experiment 2)Events 3)Sample Space 4)Mutually Exclusive 5)Collectively Exhaustive 6)Independent Events 7)Compliment 8)Joint Event

Quiz What’s the difference between Mutually Exclusive and Collectively Exhaustive?

When you estimate a probability You are estimating the probability of an EVENT occurring.

When rolling two die, the probability of rolling an 11 (Event A) is the probability that Event A occurs. It is written P(A) P(A) = probability that event A occurs

With a sample space of the toss of a fair die being S = {1, 2, 3, 4, 5, 6}

Find the probability of the following events: 1)An even number 2)A number less than or equal to 4 3)A number greater than or equal to 5.

Answers 1)P(even number) = P(2) + P(4) + P(6)= 1/6 + 1/6 + 1/6 = 3/6 =1/2 2)P(number ≤ 4) = P(1) + P(2) + P(3) + P(4)= 1/6 + 1/6 + 1/6 + 1/6 = 4/6 = 2/3 3)P(number ≥ 5) = P(5) + P(6) = 1/6 + 1/6 = 2/6 = 1/3

Approaches to Assigning Probabilities The Relative Frequency The Classical Approach The Subjective Approach

Classical Approach to Assigning Probability Probability based on prior knowledge of the process involved with each outcome equally likely to occur in the long-run if the selection process is continually repeated.

Relative Frequency (Empirical) Approach to Assigning Probability Probability of an event occurring based on observed data. By observing an experiment n times, if Event A occurs m times of the n times, the probability that A will occur in the future is P(A) = m /n

Example of Relative Frequency Approach 1000 students take a probability exam. 200 students score an A. P(A) = 200/1000 =.2 or 20%

The Relative Frequency Approach assigned probabilities to the following simple events What is the probability a student will pass the course with a C or better? P(A) =.2 P(B) =.3 P(C) =.25 P(D) =.15 P(F) =.10

Subjective Approach to Assigning Probability Probability based on individual’s past experience, personal opinion, analysis of situation. Useful if probability cannot be determined empirically.

We leave Base Camp; the Ascent for the Summit Begins!

From a survey of 200 purchasers of a laptop computer, a gender-age profile is summarized below:

These two categories (gender and age) can be summarized together in a contingency or cross-tab table which allows the viewer to see how these two categories interact

Marginal Probability The probability that any one single event will occur. Example: P(M) = 120/200 =.6

What’s the probability of being under 30? What’s the probability of being female? What’s the probability of being either under 30 or over 45?

What is the complement of being male? P(M C ) or P(M’)

Joint Probability The probability that both Events A and B will occur. This is written as P(A and B)

What is the probability of selecting a purchaser who is female and under age 30? P(F and U) = 40/200 =.2 or 20%

Probability of A or B The probability that either of two events will occur. This is written as P(A or B). Use the General Addition Rule which eliminates double-counting.

General Addition Rule P(A or B) = P(A) + P(B) – P(A and B)

What is the probability of selecting a purchaser who is male OR under 30 years of age? P(M or U) = P(M) + P(U) – P(M and U) =( – 60) / 200 = 160 / 200 =.8 or 80%

We can use raw data Northeast D Southeast E Midwest F West G Finance A Manufacturing B Communication C

Or we can convert our contingency table into percentages Northeast D Southeast E Midwest F West G Finance A Manufacturing B Communication C

P(Midwest) = ? P(C or D) = ? P(E or A) =? North east D South east E Midw est F West G Finan ce A Manuf acturi ng B Com munic ation C North east D South east E Midw est F West G Finan ce A Manuf acturi ng B Com munic ation C

Solution North east D South east E Midwe st F West G Finan ce A Manuf acturi ng B Comm unicat ion C P(F) =.21 P(C or D) = P(C) + P(D) – P(C & D) = =.64 or 64% P(E or A) = =.40 or 40%

Addition Rule for Mutually Exclusive Events: P(A or B) = P(A) + P(B)

Frequently, we need to know how two events are related.

Conditional Probability We would like to know the probability of one event occurring given the occurrence of another related event.

Conditional Probability The probability that Event A occurs GIVEN that Event B occurs. P (A | B) B is the event known to have occurred and A is the uncertain event whose probability you seek, given that Event B has occurred.

What is the probability of selecting a female purchaser given the selected individual is under 30 years of age? P(F | U) = 40 / 100 =.4 Interpretation: There is a 40% probability of selecting a female given the selected individual is under 30 years of age.

Hypoxia Question 1: How is P(F|U) different than the P(F)?

There is a 40% chance of selecting a female purchaser given no prior information about U. P(F)=.4 This means that being given the information that the person selected is under 30 has no effect on the probability that a female is selected.

In other words, U has no effect on whether F occurs. Such events are said to be INDEPENDENT Events A and B are independent if the probability of Event A is unaffected by the occurrence or non-occurence of Event B

Statistical Independence Events A and B are independent if and only if: P(A | B) = P(A) {assuming P(B) ≠ 0}, or P(B | A) = P(B) {assuming P(A) ≠ 0}, or P(A and B) = P(A) ∙ P(B).

What is the probability of selecting a female purchaser given the selected individual is between years of age? Are the events independent?

P(F | B) = 30/50 =.6 Test for independence: P(F | B) = P(F) 30/50 = 80/200.6 ≠.4 The events are not independent.

What is the rate of promotion among female assistant professors? What is the rate of promotion among male assistant professors? Is it reasonable to accuse the university of gender bias?