1 Melikyan/DM/Fall09 Discrete Mathematics Ch. 6 Counting and Probability Instructor: Hayk Melikyan Today we will review sections 6.1, 6.2, 6.3 we will start with the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.

2 Melikyan/DM/Fall09 Sample Space, Events, and PROBABILITY Department of Mathematics and CS we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.

3 Melikyan/DM/Fall09 Blaise Pascal-father of modern probability Blaise Pascal Born: 19 June 1623 in Clermont (now Clermont-Ferrand), Auvergne, France Died: 19 Aug 1662 in Paris, France In correspondence with Fermat he laid the foundation forFermat the theory of probability. This correspondence consisted oftheory of probability five letters and occurred in the summer of They considered the dice problem, already studied by Cardan,Cardan and the problem of points also considered by Cardan and,Cardan around the same time, Pacioli and Tartaglia. The dicePacioliTartaglia problem asks how many times one must throw a pair of dice before one expects a double six while the problem of points asks how to divide the stakes if a game of dice is incomplete. They solved the problem of points for a two player game but did not develop powerful enough mathematical methods to solve it for three or more players.

4 Melikyan/DM/Fall09 Pascal

5 Melikyan/DM/Fall09 Probability 1. Important in inferential statistics, a branch of statistics that relies on sample information to make decisions about a population. 2. Used to make decisions in the face of uncertainty.

6 Melikyan/DM/Fall09 Terminology 1. Random experiment : is a process or activity which produces a number of possible outcomes. The outcomes which result cannot be predicted with absolute certainty. Example 1 : Flip two coins and observe the possible outcomes of heads and tails

7 Melikyan/DM/Fall09 Examples 2. Select two marbles without replacement from a bag containing 1 white, 1 red and 2 green marbles. 3. Roll two die and observe the sum of the points on the top faces of each die. All of the above are considered experiments.

8 Melikyan/DM/Fall09 Terminolog y Sample space : is a list of all possible outcomes of the experiment. The outcomes must be mutually exclusive and exhaustive. Mutually exclusive means they are distinct and non-overlapping. Exhaustive means complete. Event: is a subset of the sample space. An event can be classified as a simple event or compound event.

9 Melikyan/DM/Fall09 Terminology 1. Select two marbles in succession without replacement from a bag containing 1 red, 1 blue and two green marbles. 2. Observe the possible sums of points on the top faces of two dice.

10 Melikyan/DM/Fall09 3. Select a card from an ordinary deck of playing cards (no jokers) The sample space would consist of the 52 cards, 13 of each suit. We have 13 clubs, 13 spades, 13 hearts and 13 diamonds. A simple event: the selected card is the two of clubs. A compound event is the selected card is red (there are 26 red cards and so there are 26 simple events comprising the compound event) 4. Select a driver randomly from all drivers in the age category of (Identify the sample space, give an example of a simple event and a compound event)

11 Melikyan/DM/Fall09 More examples Roll two dice. Describe the sample space of this event. You can use a tree diagram to determine the sample space of this experiment. There are six outcomes on the first die {1,2,3,4,5,6} and those outcomes are represented by six branches of the tree starting from the “tree trunk”. For each of these six outcomes, there are six outcomes, represented by the brown branches. By the fundamental counting principle, there are 6*6=36 outcomes. They are listed on the next slide.

12 Melikyan/DM/Fall09 Sample space of all possible outcomes when two dice are tossed. (1,1), (1,2), (1,3), (1,4), (1,5) (1,6) (2,1), (2,2), (2,3), (2,4), (2,5), (2,6) (3,1), (3,2), (3,3), (3,4), (3,5), (3,6) (4,1), (4,2), (4,3), (4,4), (4,5), (4,6) (5,1), (5,2), (5,3), (5,4), (5,5), (5,6) (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) Quite a tedious project !!

13 Melikyan/DM/Fall09 Probability of an event Definition: sum of the probabilities of the simple events that constitute the event. The theoretical probability of an event is defined as the number of ways the event can occur divided by the number of events of the sample space. Using mathematical notation, we have P(E) = n(E) is the number of ways the event can occur and n(S) represents the total number of events in the sample space.

14 Melikyan/DM/Fall09 Examples Probability of a sum of 7 when two dice are rolled. First we must calculate the number of events of the sample space. From our previous example, we know that there are 36 possible sums that can occur when two dice are rolled. Of these 36 possibilities, how many ways can a sum of seven occur? Looking back at the slide that gives the sample space we find that we can obtain a sum of seven by the outcomes { (1,6), (6,1), (2,5), (5,2), (4,3), (3,4)} There are six ways two obtain a sum of seven. The outcome (1,6) is different from (6,1) in that (1,6) means a one on the first die and a six on the second die, while a (6,1) outcome represents a six on the first die and one on the second die. The answer is P(E)= =

15 Melikyan/DM/Fall09 PROBABILITIES FOR SIMPLE EVENTS Given a sample space S = {e 1, e 2,..., e n }. To each simple event e i assign a real number denoted by P(e i ), called the PROBABILITY OF THE EVENT e i. These numbers can be assigned in an arbitrary manner provided the Following two conditions are satisfied: (a) The probability of a simple event is a number between 0 and 1, inclusive. That is, 0 ≤ p(e i ) ≤ 1 (b) The sum of the probabilities of all simple events in the sample space is 1. That is, P(e 1 ) + P(e 2 ) P(e n ) = 1 Any probability assignment that meets these two conditions is called an ACCEPTABLE PROBABILITY ASSIGNMENT.

16 Melikyan/DM/Fall09 PROBABILITY OF AN EVENT E Given an acceptable probability assignment for the simple events in a sample space S, the probability of an arbitrary event E, denoted P ( E ), is defined as follows: (a) P ( E ) = 0 if E is the empty set. (b) If E is a simple event, then P ( E ) has already been assigned. (c) If E is a compound event, then P ( E ) is the sum of the probabilities of all the simple events in E. (d) If E = S, then P ( E ) = P ( S ) = 1.

17 Melikyan/DM/Fall09 STEPS FOR FINDING THE PROBABILITY OF AN EVENT E (a) Set up an appropriate sample space S for the experiment. (b) Assign acceptable probabilities to the simple events in S. (c) To obtain the probability of an arbitrary event E, add the probabilities of the simple events in E.

18 Melikyan/DM/Fall09 EMPIRICAL PROBABILITY If an experiment is conducted n times and event E occurs with FREQUENCY f ( E ), then the ratio f ( E )/ n is called the RELATIVE FREQUENCY of the occurrence of event E in n trials. The EMPIRICAL PROBABILITY of E, denoted by P ( E ), is given by the number (if it exists) that the relative frequency f ( E )/ n approaches as n gets larger and larger. For any particular n, the relative frequency f ( E )/ n is also called the APPROXIMATE EMPIRICAL PROBABILITY of event E : (The larger n is, the better the approximation.)

19 Melikyan/DM/Fall09 PROBABILITIES UNDER AN EQUALLY LIKELY ASSUMPTION If, in a sample space S = { e 1, e 2,..., e n }, each simple event e i is as likely to occur as any other, then P ( e i ), for i = 1, 2,..., n, i. e. assign the same probability, 1/ n, to each simple event. The probability of an arbitrary event E in this case is:

20 Melikyan/DM/Fall09 Example of classical probability Example: Toss two coins. Find the probability of at least one head appearing. Solution: At least one head is interpreted as one head or two heads. Step 1: Find the sample space:{ HH, HT, TH, TT} There are four possible outcomes. Step 2: How many outcomes of the event “at least one head” Answer: 3 : { HH, HT, TH} Step 3: Use P(E)= = ¾ = 0.75 = 75%

21 Melikyan/DM/Fall09 Counting the elements of a set Theorem: If m and n are integers and m  n, then there are n – m + 1 integers from m to n inclusive. Example: a) How many two digit numbers are divisible by 7? b) What is the probability that a randomly chosen two digit number is divisible by 7

22 Melikyan/DM/Fall09 Possibility Trees and The Multiplication Rule Teams A and B are to play each other repeatedly until one wins two games in a row or a total three games. –What is the probability that five games will be needed to determine the winner?

23 Melikyan/DM/Fall09 Next Example Suppose there are 4 I/O units and 3 CPUs. In how many ways can I/Os and CPUs be attached to each other when there are no restrictions

24 Melikyan/DM/Fall09 Multiplication Rule Theorem ( The Multiplication Rule): If an operation consists of k steps each of which can be performed in n i ways (i = 1, 2, …, k), then the entire operation can be performed in n 1 n 2... n k ways. Example: A PIN number is a sequence of any four letters selected from 26 and the ten digits, with repetition allowed. a) How many PIN numbers are possible. b) Number of PIN’s if no repetition

25 Melikyan/DM/Fall09 Multiplication Rule : Example Three officers – a president, a treasurer and a secretary are to be chosen from four people: Alice, Bob, Cindy and Dan. Alice cannot be a president, Either Cindy or Dan must be a secretary. How many ways can the officers be chosen?

26 Melikyan/DM/Fall09 Example (continue) What if we reorder the steps of selections Select first the secretary, then president, and the treasurer

27 Melikyan/DM/Fall09 Permutations Definition: A permutation of a set of objects is an ordering of these objects Theorem: The number of permutations of a set of n objects is n! Consider a permutation as an n-step selection:

28 Melikyan/DM/Fall09 Example: How many ways can the letters in the word COMPUTER be arranged in a row? How many ways can the letters in the word COMPUTER be arranged if the letters CO must remain next to each other.

29 Melikyan/DM/Fall09 Permutation of Selected Objects Definition: An r-permutation of a set of n elements is an ordered selection of r elements taken from a set of n elements: P(n, r) Theorem: If n and r are integers and 1  r  n, then P(n, r) = n(n -1)(n - 2)... (n – r +1) or P(n, r) = n! / (n – r)! Show that P(n, 2) + P(n, 1) = n 2 The number of r-permutations of a set of n elements is denoted by P(n, r)

30 Melikyan/DM/Fall09 The sketch of last theorems proof: Formation of r-permutation can be thought as an r-step process: It follows from the multiplication principle that P(n, r) = n(n - 1)(n - 2)·...·(n - r + 1) = n!/(n -r)!

31 Melikyan/DM/Fall09 Addition Rule Theorem: If a finite set A is a union of k mutually disjoint sets A 1, A 2, …, A k, then n(A) = n(A 1 ) + n(A 2 ) n(A k ) Number of words of length no more than 3 Number of integers divisible by 5

32 Melikyan/DM/Fall09 Example: Counting three digit the number of integers divisible by 5

33 Melikyan/DM/Fall09 The Difference Rule Theorem: If A is a finite set and B is its subset, then n(A \ B) = n(A) – n(B) How many students are needed so that the probability of two of them having the same birthday equals 0.5? Formula for probability of the complement of an Event. If S is a finite sample space and A is an event in S then P(A C ) = 1 – P(A)

34 Melikyan/DM/Fall09 Inclusion/Exclusion Rule Theorem: If A, B, C are any finite sets, then n(A  B ) = n(A) + n(B) – n(A  B) and n(A  B  C ) = n(A) + n(B) +n(C) - – n(A  B)- n(A  C)- n(B  C) + n(A  B  C)

35 Melikyan/DM/Fall09 Examples: How many integers from 1 through 1000 are multiples of 3 or multiples of 5 How many integers from 1 through 1000 are neither multiples of 3 nor multiples of 5?

36 Melikyan/DM/Fall09 Exercises Suppose that out of 50 students, 30 know Java,18 know C++, 26 know C#, 9 know both C++ and Java, 16 know both Java and C#, 8 know C++ and C# and 47 know at least one programming language. –How many students know none of the three languages? –How many students know all three languages –How many students know exactly 2 languages?

37 Melikyan/DM/Fall09 Example (continue)

38 Melikyan/DM/Fall09 Exercises How many integers from 1 to contain the digit 6 exactly once / at least once? What is a probability that a random number from 1 to will contain two or more occurrences of digit 6? 6 new employees, 2 of whom are married are assigned 6 desks, which are lined up in a row. What is the probability that the married couple will have non-adjacent desks?

39 Melikyan/DM/Fall09 Exercises Consider strings of length n over the set {a, b, c, d}: –How many such strings contain at least one pair of consecutive characters that are the same? –If a string of length 10 is chosen at random, what is the probability that it contains at least on pair of consecutive characters that are the same? How many permutations of abcde are there in which the first character is a, b, or c and the last character is c, d, or e? How many integers from 1 through contain each of the digits 1, 2, and 3 at least once?