sample space all possible returns on the s&p500 index tomorrow, rounded to the nearest percent...
DESCRIPTION
Rules Commutative Law: Associative Law: Distributive Law:TRANSCRIPT
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Sample Space
• All possible returns on the S&P500 index tomorrow, rounded to the nearest percent– ...-4, -3, -2, ..., 3, 4, ...– This is called sample space, S
• All possible increases in the index– 1, 2, 3, ...– Subset of the sample space, E
• If return tomorrow is positive, then we’ll say the event has occurred– The outcome is part of E
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Events
• Union of two events:• Intersection of two events:• Complement of an event:• E is contained in F, or E is a subset of F:• F is a superset of E:
FE
FE
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Rules
• Commutative Law:
• Associative Law:
• Distributive Law:
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Rules Continued
• De Morgan’s laws:
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Axioms of Probability
• Consider an experiment repeated n times– n(E) = number of times an event occurs– Probability of an event = limiting frequency of
an event:
• Or, only assume that P(E) exists
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Axioms of Probability
• Probability of an event E: P(E)• Axiom 1:
• Axiom 2:
• Axiom 3
*for any sequence of mutually exclusive events
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Axioms of Probability
• What is the probability of a positive return on the S&P500 tomorrow?P({1,2,3,…})= P({1}) + P({2}) + P({3}) + …
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Propositions
• Proposition 4.1.
• Proposition 4.2.
• Proposition 4.3.
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Equally likely outcomes
• Imagine all outcomes of a probability space are equal– All outcomes are equally likely
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Counting
• If outcomes are equally likely, all you need to do is count the events in E and S
• Examples– Two dice, what’s the probability of getting a 7– Poker hand (5 cards) has 4 of a kind?
What’s this?
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Counting - Permutatinos
• How many different ordered arrangements of the letters a, b, and c are possible?
• n(n-1)(n-2).... 3 * 2 * 1 = n!
• How many different arrangements can be formed from the letters PEPPER? – Example 3d, p. 4
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Counting - Combinations
• Groups of r objects that could be formed from a total of n objects
• How many groups of 3 could be selected from A, B, C, D, and F?
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Probability and Beliefs
• I’m 60% certain that the market is going down tomorrow ....– And 40% certain that they’re going up the day
after tomorrow• What’s the probability that –Markets go up both days– Go down both days