1. Randomness and Probability
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Chapter 5
Randomness and Probability 1
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2. 5.1 Random Phenomena and Probability
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5.1 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. 2
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3. 5.1 Random Phenomena and Probability
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5.1 Random Phenomena and Probability
Sample space is a special event that is the collection of all possible outcomes.
We denote the sample space S or sometimes Ω.
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. 3
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4. 5.1 Random Phenomena and Probability
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5.1 Random Phenomena and Probability
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 value.
Empirical probability is based on repeatedly observing the event’s outcome. 4
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5. 5.2 The Nonexistent Law of Averages
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5.2 The Nonexistent Law of Averages
Many people confuse the Law of Large numbers with the so-called Law of Averages that would say that things have to even out in the short run.
The Law of Averages doesn’t exist. 5
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6. 5.3 Different Types of Probability
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5.3 Different Types of Probability
Model-Based (Theoretical) Probability
The (theoretical) probability of event A can be computed with the following equation: 6
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7. 5.3 Different Types of Probability
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5.3 Different 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. 7
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8. 5.4 Probability Rules Rule 1
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5.4 Probability Rules
Rule 1
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, 8
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9. 5.4 Probability Rules Rule 2: The Probability Assignment Rule
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5.4 Probability Rules
Rule 2: 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. 9
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10. 5.4 Probability Rules Rule 3: The Complement Rule
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5.4 Probability Rules
Rule 3: 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 A is called the “complement” of A, and is denoted AC. 10
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11. 5.4 Probability Rules Rule 4: The Multiplication Rule
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5.4 Probability Rules
Rule 4: 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. 11
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12. 5.4 Probability Rules Rule 5: The Addition Rule
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5.4 Probability Rules
Rule 5: 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 A and B are disjoint. 12
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13. 5.4 Probability Rules Rule 6: The General Addition Rule
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5.4 Probability Rules
Rule 6: 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. 13
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14. 5.5 Joint Probability and Contingency Tables
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5.5 Joint Probability and Contingency Tables
Events may be placed in a contingency table such as the one in the example below.
Example: 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: 14
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15. 5.5 Joint Probability and Contingency Tables
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5.5 Joint Probability and Contingency Tables
Marginal probability depends only on totals found in
the margins of the table. 15
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16. 5.5 Joint Probability and Contingency Tables
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5.5 Joint Probability and Contingency Tables
In the table below, the probability that a respondent chosen at random is a woman is a marginal probability.
P(woman) = 251/478 = 16
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17. 5.5 Joint Probability and Contingency Tables
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5.5 Joint Probability and Contingency Tables
Joint probabilities give the probability of two events occurring together.
P(woman and camera) = 91/478 = 17
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18. 5.5 Joint Probability and Contingency Tables
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5.5 Joint Probability and Contingency Tables
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:
P(bike|woman) = 30/251 = 18
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19. 5.6 Conditional Probability
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5.6 Conditional Probability
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. 19
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20. 5.6 Conditional Probability
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5.6 Conditional Probability
Rule 7: 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. 20
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21. 5.6 Conditional Probability
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5.6 Conditional Probability
Events A and B are independent whenever P(B|A) = P(B).
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. 21
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22. 5.7 Constructing Contingency Tables
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5.7 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. 22
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23. 5.7 Constructing Contingency Tables
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5.7 Constructing Contingency Tables
Example: 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: 23
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24. 5.7 Constructing Contingency Tables
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5.7 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. 24
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25. 5.7 Constructing Contingency Tables
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5.7 Constructing Contingency Tables
Because the cells of the table show disjoint events, the probabilities always add to the marginal totals going across rows or down columns. 25
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26. What Can Go Wrong? Beware of probabilities that don’t add up to 1.
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What Can Go Wrong?
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. 26
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27. QTM1310/ Sharpe
What Have We Learned?
Probability is based on long-run relative frequencies.
The Law of Large Numbers speaks only of long-run behavior and should not be misinterpreted as a law of averages. 27
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28. QTM1310/ Sharpe
What Have We Learned?
Some basic rules for combining probabilities of outcomes
to find probabilities of more complex events:
Probability for any event is between 0 and 1
Probability of the sample space, S, the set of possible outcomes = 1
Complement Rule
Multiplication Rule for independent events
General Addition Rule
General Multiplication Rule 28
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