lecture 6 randomness and probability. random phenomena and probability with random phenomena, we...
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- LECTURE 6 Randomness and Probability
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- RANDOM PHENOMENA AND PROBABILITY With random phenomena, we cant predict the individual outcomes, but we can hope to understand characteristics of their long-run behavior. For any random phenomenon, each attempt, or trial, generates an outcome. We use the more general term event to refer to outcomes or combinations of outcomes.
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- SAMPLE SPACES A sample space is a special event that is the collection of all possible outcomes. We denote the sample space S or sometimes (omega) The probability of an event is its long-run relative frequency. Independence means that the outcome of one trial doesnt influence or change the outcome of another.
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- LAW OF LARGE NUMBERS The Law of Large Numbers (LLN) states that if the events are independent, then as the number of trials increases, the long-run relative frequency of an event gets closer and closer to a single (true) value. Empirical probability is based on repeatedly observing the events outcome.
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- LAW OF AVERAGES Many people confuse the Law of Large Numbers with the so-called Law of Averages The Law of Averages doesnt exist. Presumably, the Law of Averages would say that things have to even out in the short run. No luck this time, more luck next time.
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- TYPES OF PROBABILITY: THEORETICAL PROBABILITY
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- TYPES OF PROBABILITY: PERSONAL PROBABILITY A subjective, or personal probability expresses your uncertainty about the outcome. Although personal probabilities may be based on experience, they are not based either on long-run relative frequencies or on equally likely events.
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- PROBABILITY RULES: RULE 1
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- PROBABILITY RULES: RULE 2
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- PROBABILITY RULES: RULE 3
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- PROBABILITY RULES: EXAMPLE Lees Lights sell lighting fixtures. Lee records the behavior of 1000 customers entering the store during one week. Of those, 300 make purchases. What is the probability that a customer doesnt make a purchase? If P( Purchase ) = 0.30, then P( no purchase ) = 1 P( Purchase ) =1 0.30 = 0.70
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- PROBABILITY RULES: RULE 4
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- PROBABILITY RULES: EXAMPLE
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- PROBABILITY RULES: RULE 5
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- PROBABILITY RULES: EXAMPLE Some customers prefer to see the merchandise but then make their purchase online. Lee determines that theres an 8% chance of a customer making a purchase in this way. We know that about 30% of customers make purchases when they enter the store. What is the probability that a customer who enters the store makes no purchase at all? P( purchase in the store or online ) = P ( purchase in store ) + P( purchase online ) = 0.30 + 0.08 = 0.38 P( no purchase ) = 1 0.38 = 0.62
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- PROBABILITY RULES: RULE 6
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- PROBABILITY RULES: EXAMPLE
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- Car Inspections You and a friend get your cars inspected. The event of your cars passing inspection is independent of your friends car. If 75% of cars pass inspection what is the probability that Your car passes inspection? Your car doesnt pass inspection? Both cars pass inspection? At least one of two cars passes? Neither car passes inspection?
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- PROBABILITY RULES: EXAMPLE
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- CONTINGENCY TABLES Events may be placed in a contingency table such as the one in the example below. As part of a Pick Your Prize Promotion, a store invited customers to choose which of three prizes theyd like to win. The responses could be placed in the following contingency table:
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- MARGINAL PROBABILITY Marginal probability depends only on totals found in the margins of the table.
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- MARGINAL PROBABILITY
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- JOINT PROBABILITIES
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- CONDITIONAL DISTRIBUTION
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- CONDITIONAL PROBABILITY
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- PROBABILITY RULES: RULE 7
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- PROBABILITY RULES: EXAMPLE
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- Are Prize preference and Sex independent? If so, P( bike|woman ) will be the same as P( bike ). Are they equal? P( bike|woman )= 30/251 = 0.12 P( bike ) = 90/478 = 0.265 0.12 0.265 Since the two probabilites are not equal, Prize preference and Sex and not independent.
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- CONSTRUCTING CONTINGENCY TABLES If youre given probabilities without a contingency table, you can often construct a simple table to correspond to the probabilities and use this table to find other probabilities.
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- CONSTRUCTING CONTINGENCY TABLES A survey classified homes into two price categories (Low and High). It also noted whether the houses had at least 2 bathrooms or not (True or False). 56% of the houses had at least 2 bathrooms, 62% of the houses were Low priced, and 22% of the houses were both. Translating the percentages to probabilities, we have:
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- CONSTRUCTING CONTINGENCY TABLES The 0.56 and 0.62 are marginal probabilities, so they go in the margins. The 22% of houses that were both Low priced and had at least 2 bathrooms is a joint probability, so it belongs in the interior of the table.
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- CONSTRUCTING CONTINGENCY TABLES The 0.56 and 0.62 are marginal probabilities, so they go in the margins. The 22% of houses that were both Low priced and had at least 2 bathrooms is a joint probability, so it belongs in the interior of the table. Because the cells of the table show disjoint events, the probabilities always add to the marginal totals going across rows or down columns.
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- Constructing Contingency Tables CONSTRUCTING CONTINGENCY TABLES: EXAMPLE A national survey indicated that 30% of adults conduct their banking online. It also found that 40% under the age of 50, and that 25% under the age of 50 and conduct their banking online. What percentage of adults do not conduct their banking online? What type of probability is the 25% mentioned above? Construct a contingency table showing joint and marginal probabilities. What is the probability that an individual who is under the age of 50 conducts banking online? Are Banking online and Age independent?
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- Constructing Contingency Tables CONSTRUCTING CONTINGENCY TABLES: EXAMPLE What percentage of adults do not conduct their banking online? 100% 30% = 70% What type of probability is the 25% mentioned above? Marginal Construct a contingency table showing joint and marginal probabilities.
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- CONSTRUCTING CONTINGENCY TABLES: EXAMPLE What is the probability that an individual who is under the age of 50 conducts banking online? 0.25/0.40 = 0.625 Are Banking online and Age independent? No. P( banking online|under 50 ) = 0.625, which is not equal to P( banking online ) = 0.30.
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- QUICK SUMMARY Beware of probabilities that dont add up to 1. Dont add probabilities of events if theyre not disjoint. Dont multiply probabilities of events if theyre not independent. Dont confuse disjoint and independent.
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- LEARNING OUTCOMES Apply the facts about probability to determine whether an assignment of probabilities is legitimate. Probability is long-run relative frequency. Individual probabilities must be between 0 and 1. The sum of probabilities assigned to all outcomes must be 1. Understand the Law of Large Numbers and that the common understanding of the Law of Averages is false.
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- LEARNING OUTCOMES Know the 7 rules of probability and how to apply them. Know how to construct and read a contingency table. Know how to define and use independence..
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