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Chapter 6 Discrete Probability

歐亞書局. Discrete Mathematics and Its Applications Sixth Edition By Kenneth Rosen. Chapter 6 Discrete Probability. 歐亞書局. 6.1 An Introduction to Discrete Probability 6.2 Probability Theory 6.3 Bayes’ Theorem 6.4 Expected Value and Variance. P. 1.

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Chapter 6 Discrete Probability

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  1. 歐亞書局 Discrete Mathematics and Its Applications Sixth Edition By Kenneth Rosen Chapter 6Discrete Probability

  2. 歐亞書局 • 6.1 An Introduction to Discrete Probability • 6.2 Probability Theory • 6.3 Bayes’ Theorem • 6.4 Expected Value and Variance P. 1

  3. 6.1 An Introduction to Discrete Probability • Gambling • Tossing a die • Finite Probability • We will restrict to experiments that have finitely many outcomes • Experiment: a procedure that yields one of a given set of possible outcomes • Sample space: the set of possible outcomes • Event: a subset of the sample space

  4. Definition 1: If S is a finite sample space of equally likely outcomes, and E is an event (a subset of S), then the probability of E is p(E)=|E|/|S|. • Ex.1-8 • Lotteries • Poker • Sampling without replacement • Sampling with replacement

  5. The probability of combinations of events • Theorem 1: Let E be an event in a sample space S. The probability of the event Ē, the complementary event of E, is given by:p(Ē) = 1 – p(E). • Proof • Ex.8

  6. Theorem 2: Let E1 and E2 be events in the sample space S. Then,p(E1E2)=p(E1)+p(E2)-p(E1E2). • Ex.9 • Probabilistic reasoning • Determining which of two events is more likely • Ex.10

  7. 6.2 Probability Thoery • Assigning probabilities • We assign a probability p(s) to each outcome s • 0<=p(s)<=1 for each sS (S: finite or countable number of outcomes) • sSp(s)=1 • For n possible outcomes x1, x2, …, xn • 0<=p(xi)<=1 for i=1,2,…,n • i=1..np(xi)=1 • P: probability distribution • Ex.1

  8. Definition 1: Suppose that S is a set with n elements. The uniform distribution assigns the probability 1/n to each element of S. • Definition 2: The probability of the event E is the sum of the probabilities of the outcomes in E. That is,p(E)= sEp(s) • Selecting at random • Ex.2

  9. Combinations of events • The same as Sec.6.1 • p(Ē) = 1 – p(E) • p(E1E2)=p(E1)+p(E2)-p(E1E2) • Theorem 1: If E1, E2, … is a sequence of pairwise disjoint events in a sample space S, thenp(iEi)=ip(Ei).

  10. Conditional Probability • Definition 3: Let E and F be events with p(F)>0. The conditional probability of E given F, denoted by p(E|F), is defined asp(E|F)=p(EF)/p(F). • Ex.3-4 • Independence • P(E|F)=p(E) • Definition 4: The events E and F are independent if and only if p(EF)=p(E)p(F) • Ex.5-7

  11. Bernoulli Trials and the Binomial Distribution • Bernoulli trial: each performance of an experiment with two possible outcomes • A success, or a failure • Ex.8 • Theorem 2: The probability of exactly k successes in n independent Bernoulli trials, with probability of success p and probability of failure q=1-p, isC(n,k)pkqn-k. • Denoted by b(k;n,p), binomial ditribution • Ex.9

  12. Random Variable • Definition 5: A random variable is a function from the sample space of an experiment to the set of real numbers. • Not a variable, not random • Ex.10 • Definition 6: The distribution of a random variable X on a sample space S is the set of pairs (r, p(X=r)) for all rS, where p(X=r) is the probability that X takes the value r. • Ex. 11-12

  13. The Birthday Problem • The smallest number of people needed in a room so that it’s more likely than not that at lease two of them have the same day of year as their birthday • Similar to hashing functions • Ex.13 (The birthday problem) • Ex.14 (Probability of a collision in hashing functions)

  14. Monte Carlo Algorithms • Deterministic vs. probabilistic algorithms • Monte Carlo algorithms • Always produce answers to problems, but a small probability remains that the answers may be incorrect • This probability decreases rapidly when the algorithm carries out sufficient computation • Ex. 15-16

  15. The Probabilistic Method • Theorem 3: (The Probabilistic Method) If the probability that an element of a set S does not have a particular property is less than 1, there exists an element in S with this property. • Nonconstructive existence proof • Theorem 4: If k is an integer with k>=2, then R(k,k)>=2k/2. • Proof

  16. 6.3 Bayes’ Theorem • Ex.1 • Theorem 1: (Bayes’ Theorem) Suppose that E and F are events from a sample space S such that p(E)0 and p(F)0. Thenp(F|E) = p(E|F)p(F)/(p(E|F)p(F)+p(E|~F)p(~F)). • Proof • Ex.2

  17. Theorem 2: (Generalized Bayes’ Theorem) Suppose that E is an event from a sample space S and F1, F2, …, Fn are mutually exclusive events such that i=1..nFi=S. Assume that p(E)0 and p(Fi)0 for i=1,2,…,n. Thenp(Fj|E) = p(E|Fj)p(Fj)/i=1..np(E|Fi)p(Fi).

  18. Bayesian Spam Filters • Bayesian spam filters • B: set of spam messages, G: set of good messages • nB(w), nG(w) • p(w)=nB(w)/|B|, q(w)= nG(w)/|G| • P(S|E) = p(E|S)p(S)/(p(E|S)p(S)+p(E|~S)p(~S)). • Assume p(S)=p(~S)=1/2 • P(S|E)=p(E|S)/(p(E|S)+p(E|~S)) • P(E|S)=p(w), p(E|~S)=q(w) • r(w)=p(w)/(p(w)+q(w)) • Ex.3-4 • r(w1,w2,…,wk)=i=1..kp(wi)/(i=1..kp(wi)+i=1..kq(wi))

  19. Thanks for Your Attention!

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