ics141-lecture26-Probability

ics141-lecture26-Probability - 26-1ICS 141: Discrete...

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Unformatted text preview: 26-1ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiICS141: Discrete Mathematics for Computer Science IDepartment of Information and Computer SciencesUniversity of HawaiiStephen Y. Itoga26-2ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiLecture 26Chapter 6. Discrete Probability6.2 Probability Theory6.3 Bayes Theorem6.4 Expected Value and Variance26-3ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiSome material in these slides were taken/adapted from the slides made by Prof. Michael P. Frank and Prof. Jonathan L. Gross which are provided through the publisher of Discrete Mathematics and Its Applications written by Kenneth H. Rosen.Some slides were done by Prof. Baek26-4ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiBernoulli TrialsEach performance of an experiment with two possible outcomes is called a Bernoulli trialOutcome:successor failureProb. of success (p) and prob. of failure (q) Then p + q = 1Many problems can be solved by determining the probability of ksuccesses when an experiment consists of nindependent Bernoulli trials Example: A coin is biased so that the probability of heads is 2/3. what is the probability that exactly four heads come up when the coin is flipped seven times, assuming that the flips are independent?218756031635)3/1()3/2)(4,7(734==C26-5ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiBinomial DistributionTheorem 2: The probability of exactly k successes in nindependent Bernoulli trials, with probability of success pand probability of failure q = 1 p, is b(k; n, p): the probability of ksuccess in nindependent Bernoulli trials with probability of success pand probability of failure q = 1 p.Considered as a function of k, we call this function thebinomial distribution.knkqpknC-),(knkqpknCpnkb-=),(),;(26-6ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiBinomial Distribution ExampleWhat is the probability that exactly eight 0 bits are generated when 10 bits are generated with the probability that a 0 bit is generated is 0.9?Binomial distribution withk= 8, n= 10, and p(probability of 0 being generated) = 0.9knkqpknCpnkb-=),(),;(1937102445.)1.()9.)(8,10()9.,10;8(28==Cb26-7ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiRandom VariablesA random variablerandom variableis a functionfrom the sample space of an experiment to the set of real numbers. That is, a random variable assigns a real number to each possible outcome.Note that a random variable is a function and not random!Example: suppose that a coin is flipped three times. Let X(t) be the random variable that equals the number of heads that appear when tis the outcome....
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This note was uploaded on 09/12/2008 for the course ICS 141 taught by Professor Idk during the Fall '08 term at Hawaii.

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ics141-lecture26-Probability - 26-1ICS 141: Discrete...

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