461191 Discrete Math Lecture 6: Discrete Probability San Ratanasanya CS, KMUTNB

Today’s topics Basic of Counting Review Administrivia Probability Baye’s Theorem Expected Value and Variance

Basic Counting Tools Basic rules of counting Sum rule: either task 1 or task 2 have to be done Product rule: both task 1 and task 2 have to be done Task 1 and Task 2 are independent Inclusion and Exclusion Principle Exclude one that count more than once or irrelevant count Include one that miss Pigeonhole Principle If there are k nest for at least k+1 pigeons, then there must be at least one nest that has more than one pigeons

Example How many strings of five ASCII character contain the ‘@’ character at least once? (ASCII has 128 characters) 1285 - 1275

Example A computer company receives 350 applications from computer graduates for a job planning a line of new Web servers. Suppose that 220 of these people majored in CS, 147 majored in business, and 51 majored both in CS and business. How many of these applications majored in CS nor in business? |A  B| = 220 + 147 - 51 = 316  U - |A  B| = 350 – 316 = 34

Example g pigeonhole A drawer contains a dozen brown socks and a dozen black socks, all un matched. A man takes socks out at random in the dark. How many socks must he take out to be sure that he has at least two socks of the same color? How many socks must he take out to be sure that he has at least two black socks? 3 14

Binomial Coefficient THEOREM 1: The Binomial Theorem Let x,y,n be positive integer THEOREM 2: PASCAL’S IDENTITY Let n and k be positive integer with n > k then THEOREM 3: Let n be positive integer, then

Example How many terms are therein the expansion of (x + y)15? What is the difference between the coefficient of the term x8y3 and x5y6? 215 |C(11, 8) – C(11, 5)| = |165 – 462| = 297

Permutations and Combinations Ordered arrangement of a set of distinct objects. An ordered arrangement of r elements of a set is r-permutation Combinations An unordered selection of r elements from the set is r-combination P(n, r) = n! (n - r)! C(n, r) = P(n, r) = n! r! r!(n – r)!

Example A group contain n men and n women. How many ways are there to arrange these people in a row if the men and women alternate? In how many ways can a set of five letters be selected from the English alphabet? 2(n!)2 C(26, 5) = 65780

Generalized Permutations and Combinations Permutations and Combinations with repetition Permutations with Indistinguishable Objects Distributing Objects into Boxes (4 cases) Distinguishable objects and distinguishable boxes Distinguishable objects and indistinguishable boxes Indistinguishable objects and distinguishable boxes Indistinguishable objects and indistinguishable boxes

Example General PC How many string of six letter are there? How many ways are there to select three unordered elements from a set with five elements when repetition is allowed? 266 C(5+3-1, 3) = C(7, 3) = 35

Generating Permutations and Combinations Counting permutations or combinations can solve many problems Counting is sometimes not enough. We need to generate either one of them to find the solution.

Example Generate PC What is the next permutation in lexicographic order after 362541? Count number of permutations = n! -1 = 6! – 1 = 719 Lexicogrphic: aj < aj+1. Therefore, the next number is 364125 (Kakuro) Let S = {1, 2, …, 9}. What are the subset that has 3 elements of S and their sum is 15? Count number of combinations = 9! / 3!6! = 84 Generate subset: {1, 5, 9}, {1, 6, 8}, {2, 4, 9}, {2, 5, 8} ,{2, 6, 7}, {3, 4, 8}, {3, 5, 7}, {4, 5, 6}

Administrivia Midterm Exam is in next 2 weeks. Please be prepared! All assignments MUST be submitted before Midterm.

Discrete Probability and Probability Theory

Finite Probability Laplace’s definitions: Experiment – a procedure that yields one of a given set of possible outcomes. Sample space – the set of possible outcomes from the experiment. Event – a subset of the sample space. Definition 1: If S is a finite sample space of equally likely outcomes, and E is an event, that is , a subset of S, then the probability of E is p(E) = |E| |S|

The Probability of Combinations of Events Theorem 1: Let E be an event in a sample space S. The probability of the event E, the complementary event of E, is given by p(E) = 1 – p(E) Theorem 2: Let E1 and E2 be events in sample space S. Then p(E1  E2) = p(E1) + p(E2) – p(E1  E2)

Example There are many lotteries now that award enormous prizes to people who correctly choose a set of six numbers out of the first n positive integers, where n is usually between 30 and 60. What is the probability that a person picks the correct numbers out of 40? What is the probability that a positive integer selected at random from the set of positive integers not exceeding 100 is divisible by either 2 or 5? 1 / C(40, 6) = 1 / 3838380 = 0.00000026 0.5 + 0.2 – 0.1 = 0.6

Probabilistic Reasoning The Monty Hall Three-door Puzzle You are asked to select one of the 3 doors to win the price. Once you select the door, the host, who knows what is behind each door, he opens one of the other two doors that he know is a losing door. Then he ask you whether you would like to switch doors. Should you change doors or keep your original selection? You should change the door whenever you have chance! The probability of wining will change from 1/3 to 2/3

Assigning Probabilities Let S be the sample space of an experiment with a finite or countable number of outcomes. We assign a probaility p(s) to each outcome s such that these 2 conditions must be met 0  p(s)  1 for each s  S The function p() is called a probability distribution

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, (Note that when E is an infinite set, is a convergent infinite series.)

Example Suppose that a die is biased so that 3 appears twice as often as each number but that the other five outcomes are equally likely. What is the probability that an odd number appears when we roll this die? Let E = {1, 3, 5} p(1) = p(2) = p(4) = p(5) = p(6) = 1/7; p(3) = 2/7 Then p(E) = 4/7

Conditional Probability and Independence 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 as p(E | F) = p(E  F) p(F) Definition 4: The events E and F are independent if and only if p(E  F) = p(E)∙p(F)

Example What is the conditional probability that a family with two kids has two boys, given they have at least one boy? Are the events E, that a family with three kids has of both sexes, and F, that this family has at most one boy, independent? Let E = {BB}, F = {BB, GB, BG} then E  F = {BB} p(E|F) = (1/4)/(3/4) = 1/3 Yes. Because p(E  F) = p(E)p(F) = 3/8

Bernoulli Trials and the Binomial Distribution An experiment with only 2 outcomes either Success or Failure is called Bernoulli Trials. Let p and q be a probability of success and failure, respectively. Then, it follows that p + q = 1 Bernoulli Trails are mutually independent. 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, is b(k; n, p) = C(n, k)pkqn-k If we consider it as a function of k, we call this function the Binomial Distribution.

Random variable Definition 5: A random variable is a function fro 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. Definition 6: The distribution of random variable X on a sample space S is the set of pairs (r, p(X = r)) for all r  X(S), where p(X = r) is the probability that X takes the value r. A distribution is usually described by specifying p(X = r) fro each r  X(S).

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 t is the outcome. Find X(t). X(HHH) = 3, X(HHT) = X(HTH) = X(THH) = 2, X(TTH) = X(THT) = X(HTT) = 1, X(TTT) = 0 Because each of the eight possible outcomes Has probability of 1/8, the distribution of X(t) is given by P(X=3) = 1/8, P(X=2)=3/8, P(X=1) = 3/8, P(X=0) = 1/8

More Examples Birthday Problem. What is the minimum number of people who need to be in a room so that the probability that at least two of them have the same birthday is greater than 1/2? Example 14 - 16 1- pn = 1- 365 364 363 … 367-n 366 366 366 366 n = 22

Monte Carlo Algorithms Deterministic Algorithms Always proceed in the same way whenever given the same input. Probabilistic Algorithms Make random choices at one or more step for its efficiency in a huge number of possible case. Mote Carlo Algorithms The probability that the algorithm give the correct answer increases as more test carried out. Details in Section 6.2 page 411

The Probabilistic Method is used as existence proof to proof results about set S. See Theorem 4 for example Theorem 3: 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.

Baye’s Theorem

Bayes’ Theorem The probability that a particular event occurs on the basis of partial evidence, i.e., it is like conditional probability Theorem 1: Suppose that E and F are events from a sample space S such that p(E)  0 and p(F)  0. Then P(F | E) = p(E | F)p(F) p(E | F)p(F) + p(E | F)p(F) Theorem 2: Suppose that E is an event from a sample space S and that F1, F2, …, Fn are mutually exclusive events such that Assume that p(E)  0 and p(Fi)  0 for i = 1, 2, .., n. Then

Example We have 2 boxes. The first box contains two green balls and seven red balls; the second box contains four green balls and three red balls. Bob selects a ball by first choosing one of the two boxes at random. He then selects one of the balls in this box at random. If Bob has selected a red ball, what is the probability that he selected a ball from the first box? Details in Section 6.3 on page 417

Example: Bayesian Spam Filter Uses information about previously seen e-mail messages whether an incoming e-mail message is spam. Look for occurrences of particular word in messages. See details in Section 6.3 on page 421

Expected Value and Variance

Expected Value (of Random Variable) is a weighted average of the values of a random variable. provides central point for the distribution of values of the random variable. Definition 1: The expected value (or expectation) of the random variable X(s) on the sample space S is equal to Theorem 1: If X is a random variable and p(X = r) is the probability that X = r, so that p(X = r) = , then

Variance Definition 4: Let X be a random variable on a sample space S. The variance of X, denoted by V(X), is The standard deviation of X is defined to be Theorem 6: If X is a random variable on sample space S, then V(X) = E(X2) – E(X)2

Average-case Computational Complexity See Example 8-9 on page 431-432

Homework Section 6.1 8, 20, 35, 38, 41 Section 6.2 18, 22, 36, 37, 39, 40 Section 6.3 5, 10, 15, 18, 23 Section 6.4 2, 7, 9, 10, 22, 23 Supplementary 2, 3, 15, 16, 21, 22