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probability

# probability - 1 Probability 1.1 Motivation Life is full of...

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Unformatted text preview: 1 Probability 1.1 Motivation Life is full of uncertainty and probability is the best way we currently have to quantify it. Applications of probability arise everywhere: • Should you guess in a multiple-choice test with ve choices? What if you're not penalized for guessing? What if you're penalized 1 / 4 for every wrong answer? What if you can eliminate two of the ve possibilities? • Suppose that an AIDS test guarantees 99% accuracy: of every 100 people who have AIDS, the test returns positive 99 times (very few false negative ); of every 100 people who don't have AIDS, the test returns negative 99 times (very few false positives ) Suppose you take the test just for fun and it returns positive. How likely are you to have AIDS? Hint: the probability is not . 99 • How do you compute the average-case running time of an algorithm? • Is it worth buying a \$1 lottery ticket? Probability isn't enough to answer this question 1.2 Formalizing Probability We typically use probability to model an indeterministic experiment with a given set of possible outcomes called sample points . The sample space , typically denoted by Ω , is the set of all sample points. An event is a subset of the sample space 1 . It is the events that we will assign probability to: a number that measures how likely is that set of possible outcomes. We can study probability theory as an abstract mathematical theory but as com- puter scientists we are typically more interested in applications of this theory. This requires modelling the experiments we face and the rst step is typically deciding what the sample space should be. It is not always unambiguous as the next examples show. Example 1: We ip a coin. What is the sample space? 1 In general not every subset of Ω has to, or can, be an event. 1 • Most obvious choice: Ω = { 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 together. 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 are a doctor examining a seriously ill patient, trying to de- termine the probability that he has cancer. What is the sample space? 1.3 Probability Measures A probability measure is a function from the set of all events to the interval [0 , 1] . Intuitively, the number measures how likely that event is: probability 1 says it is certain to happen whereas probability 0 says it is certain not to happen. Probability is not an arbitrary such function: the probability that a rolled die shows 1, 2, or 3 should be the sum of the probability that it shows 1 and the probability that it shows 2 or 3.We summarize all these requirements below: Def. A probability measure P is a function mapping subsets of Ω to real num- bers such that:...
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