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Discrete Mathematics CS 2610

Discrete Mathematics CS 2610. November 5, 2008. Probability Theory. When dealing with experiments for which there are multiple outcomes- x 1 , x 2 , …, x n –we require 0  p( x i )  1 for i = 1, 2, …, n and (i=1, n) p( x i ) = 1

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Discrete Mathematics CS 2610

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  1. Discrete Mathematics CS 2610 November 5, 2008

  2. Probability Theory • When dealing with experiments for which there are multiple outcomes- x1, x2, …, xn –we require • 0  p(xi)  1 for i = 1, 2, …, n and • (i=1, n) p(xi) = 1 • We can treat p as a function that maps elements from the sample space to real values in the range [0,1]. We call such a function a probability distribution.

  3. Probability Theory Uniform Probability Distribution: p(xi) = 1/n, for i = 1, 2, …, n All outcomes are equally probable.

  4. Probability Theory Note that sum and product rules apply when dealing with probabilities too! Sequences of events are products Either/or requires sum rule and subtraction principle Complementary rule works too!

  5. Conditional Probability The conditional probability of E given F is P(E | F) = p(E  F) / p(F) This is the probability that E will/has occurred if we know that F has/will occur.

  6. Independence Two events, E and F, are independent iff p(E1  E2) = p(E1) p(E2) The two events don’t influence one another!

  7. Repeated trials If there are a number of trials being conducted, each of which has a probability of success of p and a probability of failure of q = 1 – p, then the probability of exactly k successes in n independent trials is C(n,k)pkqn-k This is called the binomial distribution.

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