Lect06_Slides

# 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 f robabilities We will therefore review the basic rules of 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. situations where events are uncertain a robability In situations where events are uncertain, a probability measures the likelihood that a particular event (or set of events) occurs. – e.g. The probability that a roll of a die comes up 6. – The probability that two randomly chosen cards add up to 1 (Bl kj k) 21 (Blackjack).

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ample Space or Universe Sample Space or Universe • Let S denote a set (collection or listing) of all possible 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 1 , s 2 ,...,s n-1 ,s n }; number of n units sold or n offers received. = [ 0 , ); stock price or salary offer. (continuous positive set space).
vents Events • An event is a collection of those states s that result in the occurrence of the event. n event can be that state s occurs or that An event can be that state s occurs or that multiple states occur, or that one of several ates occurs (there are other possibilities). states occurs (there are other possibilities). • Event A is a subset of S, denoted as A S. t A i f t h t t t i • 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 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). A 1 A 2 S
robability Probability • To each uncertain event A, or set of events, e.g. A 1 r A we would like to assign weights which or A 2 , we would like to assign weights which measure the likelihood or importance of the events a proportionate manner. in a proportionate manner. • Let P(A i ) be the probability of A i . e further assume that: We further assume that: all i i S A 1 ) ( all i i A P . 0 ) ( i A P

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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), so that A B , use the modified addition rule : P(AUB) = P(A) + P(B) –P(A B)
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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Lect06_Slides - Probability Expected Payoffs and Expected...

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