CS 70 Lecture 11

CS 70 Lecture 11 - CS 70 Spring 2010 Discrete Mathematics...

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CS 70 Discrete Mathematics and Probability Theory Spring 2010 Alistair Sinclair Note 11 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 this randomized primality testing algorithm outputs “prime” when the input is not prime is at most one in a trillion. 3. In this load-balancing scheme, the probability that any processor has to deal with more than 12 re- quests is negligible. 4. The average time between system failures is about 3 days. 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 that 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 exactly two 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 two 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, Spring 2010, Note 11 1
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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 . ω Ω Pr [ ω ] = 1, i.e., the sum of the probabilities of all outcomes is 1. The easiest way to assign probabilities to sample points is uniformly (as we saw earlier in the coin tossing example): if | Ω | = N , then Pr [ x ] = 1 N x Ω . We will see examples of non-uniform probability distributions soon.
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