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

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

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**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 measurethe 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**• Experiment • Events • Sample Space • Mutually Exclusive • Collectively Exhaustive • Independent Events • Compliment • 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**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: Thecompliment of being male is being female.**Joint Event**Has two or more characteristics.**Probability Vocabulary**• Experiment • Events • Sample Space • Mutually Exclusive • Collectively Exhaustive • Independent Events • Compliment • 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:**• An even number • A number less than or equal to 4 • 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 onprior knowledgeof the process involved with eachoutcomeequally likely to occurinthelong-runif the selection process is continually repeated.**Relative Frequency (Empirical) Approach to Assigning**Probability Probability of an event occurring based onobserved 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 isP(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 onindividual’s past experience, personal opinion, analysis ofsituation.Useful if probability cannot be determined empirically.**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**summarizedtogether in a contingency or cross-tab table which allows the viewer to see how these two categories interact**Marginal Probability**The probability that any onesingleeventwilloccur. 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?**Joint Probability**The probability thatboth EventsA and B willoccur. 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 thateitherof twoeventswilloccur. This is written asP(A or B). Use theGeneral Addition Rule which eliminatesdouble-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) =(120 + 100 – 60) / 200 = 160 / 200 = .8 or 80%