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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.
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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%
