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n10 - CS 70-2 Discrete Mathematics and Probability Theory...

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Unformatted text preview: CS 70-2 Discrete Mathematics and Probability Theory Spring 2009 Alistair Sinclair, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling — analyzing card games, dice, roulette wheels. Today it is an essential tool in engineering and the sciences. No less so in computer science, where its use is widespread in algorithms, systems, learning theory and artificial intelligence. Here are some typical statements that you might see concerning probability: 1. The chance of getting a flush in a 5-card poker hand is about 2 in 1000. 2. The chance that a particular implementation of the primality testing algorithm outputs prime when the input is composite is at most one in a trillion. 3. The average time between system failures is about 3 days. 4. In this load-balancing scheme, the probability that any processor has to deal with more than 12 re- quests is negligible. 5. There is a 30% chance of a magnitude 8.0 earthquake in Northern California before 2030. Implicit in all such statements is the notion of an underlying probability space . This may be the result of a random experiment that we have ourselves constructed (as in 1, 2 and 3 above), or some model we build of the real world (as in 4 and 5 above). None of these statements makes sense unless we specify the probability space we are talking about: for this reason, statements like 5 (which are typically made without this context) are almost content-free. Let us try to understand all this more clearly. The first important notion here is one of a random experiment. An example of such an experiment is tossing a coin 4 times, or dealing a poker hand. In the first case an outcome of the experiment might be HTHT or it might be HHHT . The question we are interested in might be “what is the chance that there are 2 H ’s?” Well, the number of outcomes that meet this condition is ( 4 2 ) = 4! 2!2! = 6 (corresponding to choosing the positions of the two H ’s in the sequence of four tosses); these outcomes are HHTT , HTHT , HTTH , THHT , THTH , TTHH . On the other hand, the total number of distinct outcomes for this experiment is 2 4 = 16. If the coin is fair then all these 16 outcomes are equally likely, so the chance that there are exactly 2 H ’s is 6 / 16 = 3 / 8. Now some terminology. The outcome of a random experiment is called a sample point . Thus HTHT is an example of a sample point. The sample space , often denoted by Ω , is the set of all possible outcomes. In our example the sample space has 16 elements: CS 70-2, Spring 2009, Note 10 1 A probability space is a sample space Ω , together with a probability Pr [ ω ] for each sample point ω , such that • 0 ≤ Pr [ ω ] ≤ 1 for all ω ∈ Ω ....
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n10 - CS 70-2 Discrete Mathematics and Probability Theory...

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