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280wk9_x4 - Formalizing Probability What do we assign...

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Formalizing Probability What do we assign probability to? Intuitively, we assign them to possible events (things that might happen, outcomes of an experiment) Formally, we take a sample space to be a set . Intuitively, the sample space is the set of possible out- comes, or possible ways the world could be. An event is a subset of a sample space. We assign probability to events: that is, to subsets of a sample space. Sometimes the hardest thing to do in a problem is to decide what the sample space should be. There’s often more than one choice A good thing to do is to try to choose the sample space so that all outcomes (i.e., elements) are equally likely This is not always possible or reasonable 1 Choosing the Sample Space Example 1: We toss a coin. What’s the sample space? Most obvious choice: { heads, tails } Should we bother to model the possibility that the coin lands on edge? What about the possibility that somebody snatches the coin before it lands? What if the coin is biased? Example 2: We toss a die. What’s the sample space? Example 3: Two distinguishable dice are tossed to- gether. What’s the sample space? (1,1), (1,2), (1,3), . . . , (6,1), (6,2), . . . , (6,6) What if the dice are indistinguishable? Example 4: You’re a doctor examining a seriously ill patient, trying to determine the probability that he has cancer. What’s the sample space? Example 5: You’re an insurance company trying to insure a nuclear power plant. What’s the sample space? 2 Probability Measures A probability measure assigns a real number between 0 and 1 to every subset of (event in) a sample space. Intuitively, the number measures how likely that event is. Probability 1 says it’s certain to happen; probability 0 says it’s certain not to happen Probability acts like a weight or measure . The prob- ability of separate things (i.e., disjoint sets) adds up. Formally, a probability measure Pr on S is a function mapping subsets of S to real numbers such that: 1. For all A S , we have 0 Pr( A ) 1 2. Pr( ) = 0; Pr( S ) = 1 3. If A and B are disjoint subsets of S (i.e., A B = ), then Pr( A B ) = Pr( A ) + Pr( B ). It follows by induction that if A 1 , . . . , A k are pairwise disjoint, then Pr( k i =1 A i ) = Σ k i Pr( A i ) . This is called finite additivity ; it’s actually more stan- dard to assume a countable version of this, called countable additivity 3 In particular, this means that if A = { e 1 , . . . , e k } , then Pr( A ) = k X i =1 Pr( e i ) . In finite spaces, the probability of a set is determined by the probability of its elements. 4
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Equiprobable Measures Suppose S has n elements, and we want Pr to make each element equally likely. Then each element gets probability 1 /n Pr( A ) = | A | /n In this case, Pr is called an equiprobable measure . Example 1: In the coin example, if you think the coin is fair, and the only outcomes are heads and tails, then we can take S = { heads,tails } , and Pr(heads) = Pr(tails) = 1/2.
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