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LECTURE 6
Randomness and Probability
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RANDOM PHENOMENA AND PROBABILITY
• With random phenomena, we can’t 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 Ω (omega)
• The probability of an event is its long-run relative frequency.
• Independence means that the outcome of one trial doesn’t 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 event’s 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 doesn’t 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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• The (theoretical) probability of event A occurring can be computed with the following equation:
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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• If the probability of an event occurring is 0, the event can’t occur.
• If the probability is 1, the event always occurs.
• For any event A, also written as
PROBABILITY RULES:RULE 1
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• The Probability Assignment Rule
• The probability of the set of all possible outcomes must be 1.
• where S represents the set of all possible outcomes and is called the sample space.
PROBABILITY RULES:RULE 2
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• The Complement Rule
• The probability of an event occurring is 1 minus the probability that it doesn’t occur.
• where the set of outcomes that are not in event is called the “complement” of , and is denoted .
PROBABILITY RULES:RULE 3
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PROBABILITY RULES:EXAMPLE
• Lee’s 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 doesn’t make a purchase?
• If P(Purchase) = 0.30, then
• P(no purchase) = 1 – P(Purchase) =1 – 0.30 = 0.70
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• The Multiplication Rule
• For two independent events A and B, the probability that both, A and B, occur is the product of the probabilities of the two events.
• provided that A and B are independent.
PROBABILITY RULES:RULE 4
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• Whether or not a caller qualifies for a platinum credit card is a random outcome. Suppose the probability of qualifying is 0.35. What is the chance that the next two callers qualify?
• Since the two different callers are independent, then
P(customer 1 qualifies and customer 2 qualifies)
= P(customer 1 qualifies) x P(customer 2 qualifies)
= 0.35 x 0.35
= 0.1225
PROBABILITY RULES:EXAMPLE
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• The Addition Rule
• Two events are disjoint (or mutually exclusive) if they have no outcomes in common.
• The Addition Rule allows us to add the probabilities of disjoint events to get the probability that either event occurs.
• where and are disjoint.
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 there’s 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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• The General Addition Rule
• The General Addition Rule calculates the probability that either of two events occurs. It does not require that the events be disjoint.
• Recall
PROBABILITY RULES:RULE 6
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PROBABILITY RULES:EXAMPLE
• Lee notices that when two customers enter the store together, their behavior isn’t independent. In fact, there’s a 20% chance they’ll both make a purchase. When two customers enter the store together, what is the probability that at least one of them will make a purchase?
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PROBABILITY RULES:EXAMPLE
• Car Inspections
• You and a friend get your cars inspected. The event of your car’s passing inspection is independent of your friend’s car. If 75% of cars pass inspection what is the probability that
• Your car passes inspection?
• Your car doesn’t pass inspection?
• Both cars pass inspection?
• At least one of two cars passes?
• Neither car passes inspection?
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PROBABILITY RULES: EXAMPLE• You and a friend get your cars inspected. The event of your
car’s passing inspection is independent of your friend’s car. If 75% of cars pass inspection what is the probability that
• Your car passes inspection?
• Your car doesn’t pass inspection?
• Both cars pass inspection?
• At least one of two cars passes?
1 – (0.25)2 = 0.9375 OR0.75 + 0.75 – 0.5625 = 0.9375
• Neither car passes inspection?
1 – 0.9375 = 0.0625
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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 they’d 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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• In the table below, the probability that a respondent chosen at random is a woman has a marginal probability of
MARGINAL PROBABILITY
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JOINT PROBABILITIES
• Joint probabilities give the probability of two events occurring together.
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• Each row or column shows a conditional distribution given one event.
• In the table above, the probability that a selected customer wants a bike, given that we have selected a woman is:
CONDITIONAL DISTRIBUTION
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• In general, when we want the probability of an event from a conditional distribution, we write P(B|A) and pronounce it “the probability of B given A.”
• A probability that takes into account a given condition is called a conditional probability.
CONDITIONAL PROBABILITY
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• The General Multiplication Rule
• The General Multiplication Rule calculates the probability that both of two events occurs. It does not require that the events be independent.
PROBABILITY RULES:RULE 7
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• What is the probability that a randomly selected customer wants a bike if the customer selected is a woman?
PROBABILITY RULES:EXAMPLE
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PROBABILITY RULES:EXAMPLE
• 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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INDEPENDENCE
• Events A and B are independent whenever
• Independent vs. Disjoint
• For all practical purposes, disjoint events cannot be independent.
• Don’t make the mistake of treating disjoint events as if they were independent and applying the Multiplication Rule for independent events.
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CONSTRUCTING CONTINGENCY TABLES
• If you’re 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 don’t add up to 1.
• Don’t add probabilities of events if they’re not disjoint.
• Don’t multiply probabilities of events if they’re not independent.
• Don’t 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. .