Lect06_Slides - Probability Expected Payoffs and Expected...

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Probability, Expected Payoffs and Expected Utility In thinking about mixed strategies, we will need to make use of probabilities . We will therefore review the basic rules of probability and then derive the notion of expected value. We will also develop the notion of expected utility as an alternative to expected payoffs . Probabilistic analysis arises when we face uncertainty. In situations where outcomes (or “states of the world”) are uncertain, a probability measures the likelihood that a particular outcome (or set of outcomes) occurs. e.g. The probability that the roll of a die comes up 6. (1/6). The probability that two randomly chosen cards in a 52 card deck add up to 21 (Natural Blackjack) (<5%).
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Sample Space or Universe Let S denote a set (collection or listing) of all possible outcomes or “states” of the environment known as the sample space or universe; a typical state is denoted as s. For example: S={s 1 , s 2 }; success/failure, or low/high price. S={s 1 , s 2 ,...,s n-1 ,s n }; number of n units sold or n offers received. S=[0, ); stock price or salary offer. (continuous positive sample space).
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States and Events A state is the result of an experiment or other situation involving uncertainty. An event is a collection of those states, s, that result in the occurrence of the event. An event can be that state s occurs or that multiple states occur, or that one of several states occurs (there are other possibilities). Event A is a subset of S, denoted as A S. Event A occurs if the true state s is an element of the set A, written as s A.
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Venn Diagrams S A 1 A 2 Illustrates the sample space and events. S is the sample space and A 1 and A 2 are events within S. “Event A 1 does not occur.” Denoted A 1 c (Complement of A 1 ) “Event A 1 or A 2 occurs.” Denoted A 1 A 2 (For probability use Addition Rules) “Event A 1 and A 2 both occur”, denoted A 1 A 2 (For probability use Multiplication Rules).
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Probability To each uncertain event A, or set of events, e.g., A 1 or A 2 , we would like to assign weights which measure the likelihood or importance of the events in a proportionate manner. Let P(A i ) be the probability of A i . We further assume that: . 0 ) ( 1 ) ( all all i i i i i A P A P S A
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Addition Rules The probability of event A or event B: P(A B) If the events do not overlap, i.e. the events are disjoint subsets of S, so that A B= , then the probability of A or B is simply the sum of the two probabilities. P(A B) = P(A) + P(B). If the events overlap, (are not disjoint) A B use the modified addition rule : P(A B) = P(A) + P(B) P(A B)
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Example Using the Addition Rule Suppose you throw two dice. There are 6x6=36 possible ways in which both can land.
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