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Discrete Mathematics PROBABILITY-1 Adil M. Khan Professor of Computer Science Innopolis University “Information: The Negative Reciprocal Value of Probability!” - Claude Shannon -

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Probability---Introduction One of the most important disciplines in Computer Science (CS). Algorithm Design and Game Theory Information Theory Signal Processing Cryptography

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Probability---Introduction---Cont. But it is also probably the least well understood Human intuition and Random events Goal: To try our best to teach you how to easily and confidently solve problems involving probability “What is the probability that … ?”

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Probability Contents Basic definitions and an elementary 4-step process Counting Conditional probability and the concept of independence Random Variable Expected value and Standard Deviation

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Probability Let’s Make a Deal The famous game show (you might have seen this problem in your books) Participant is given a choice of three doors. Behind one door is a car, behind the others, useless stuff. The participant picks a door (say door 1). The host, who knows what is behind the doors, opens another door (say door 3) which has the useless stuff. He then asks the participant if he would like to switch (pick door 2)? Is it to participant’s advantage to switch or not?

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Probability Precise Description The car is equally likely to be hidden behind the three doors.

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Probability Precise Description The car is equally likely to be hidden behind the three doors. The player is equally likely to pick each of the doors. After the player picks a door, the host must open a different door (with the useless thing behind it) and offer the player to switch. When a host has a choice of which door to pick, he is equally likely to pick each of them. Now here comes the question: “What is the probability that a player who switches wins the car?”

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Probability Solving Problems Involving Probability Model the situation mathematically Solve the resulting mathematical problem

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Probability Solving Problems Involving Probability Step 1: Finding the sample space Set of all possible outcomes of a random process

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Probability Solving Problems Involving Probability Step 1: Finding the sample space Set of all possible outcomes of a random process

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Probability Solving Problems Involving Probability Step 1: Finding the sample space Set of all possible outcomes of a random process To find this, we must understand the quantities involve in the random process Quantities in the above problem: The door concealing the car The door initially chosen by the player The door that host opens to reveal the useless thing

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Probability Finding the Sample Space Every possible value of these quantities is called an outcome. And (as said earlier) the set of all possible outcomes is called the sample space

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Probability Finding the Sample Space Every possible value of these quantities is called an outcome. And (as said earlier) the set of all possible outcomes is called the sample space A tree structure (Possibility tree) is a useful tool for keeping track of all outcomes When the number of possible outcomes is not too large

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Probability Possibility Tree The first quantity in our example is the door concealing the car Represent this as a root of tree with three branches (three doors)

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Probability Possibility Tree --- Cont. The car can be at any of these three locations

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Probability Possibility Tree --- Cont. The car can be at any of these three locations For each possible location of the car, the player can choose any of the three doors

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Probability Possibility Tree --- Cont. The car can be at any of these three locations For each possible location of the car, the player can choose any of the three doors Then the final possibility is regarding the host opening a door to reveal the useless thing Overall tree turns out to be

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Probability Possibility Tree --- Cont.

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Probability Finding The Sample Space The leaves of the possibility tree represent the outcomes of a random process The set of all leaves represent the sample space

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Probability Finding The Sample Space In our example, if we represent the leaves as a sequence of “labels” of intermediate nodes including the leaf node then,

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Probability Solving Problems Involving Probability Step 2: Defining the Events of Interest:

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Probability Solving Problems Involving Probability Step 2: Defining the Events of Interest: Remember, we want to answer the questions of type: “What is the probability that … ?”

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Probability Solving Problems Involving Probability Step 2: Defining the Events of Interest: Remember, we want to answer the questions of type: “What is the probability that … ?” Replacing the “…” with some specific event. For example, “What is the probability that the car is behind door C?” Doing this reduces S to some specific outcomes, called event of interest.

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Probability Event of Interest For the event, “What is the probability that the car is behind door C?” The set of possible outcomes reduces to

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Probability Event of Interest For the event, “What is the probability that the car is behind door C?” The set of possible outcomes reduces to Simply speaking, an event is a subset of S

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Probability Solving Problems Involving Probability Coming back to our example We want to know: “What is the probability that the player will win by switching?” This event can be represented as the following set

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Probability Solving Problems Involving Probability---Cont. Notice: Half of the outcomes are checked. Does this mean that the player wins by switching in half of all outcomes?

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Probability Solving Problems Involving Probability---Cont. Step 3: Determining Outcome Probability Assign Edge Probabilities Compute Outcome Probabilities

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Probability Equally likely probability formula E: the equally likely event S: the sample space

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Probability Solving Problems Involving Probability---Cont. Step 3: Determining Outcome Probability Assign Edge Probabilities Compute Outcome Probabilities

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Probability Edge Probabilities To understand, let’s analyze the path leading to the leaf node (A, A, B)!

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Probability Multiplication Rule The probability that Events A and B both occur is equal to the probability that Event A occurs times the probability that Event B occurs, given that A has occurred.

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Probability Outcome Probabilities To understand, let’s analyze the probability of the outcome (A, A, B).

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Probability Solving Problems Involving Probability---Cont. Step 4: Compute Event Probability

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Probability Summary To solve problems involving probability, that is, “what is the probability that … ?” Perform the following four steps: Find the sample space Define event of interest Compute outcome probabilities Compute event probability

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Probability Uniform Sample Space Strange Dice If we picked dices (a) and (b), rolled them once, what is the probability that (a) beats (b) (has a higher value)?

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Probability Applying Four-Step Method When the probability of every outcome is the same, we say such a sample space is uniform

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Probability Applying Four-Step Method Example--- Cont. So what is the probability that (a) beats (b)? Which in this case = (a) Beats (b) more than half of the time.

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Probability Applying Four-Step Method Example--- Cont. What about the following: (a) vs. (c) (b) vs. (c) Homework!

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Set Theory and Probability Sample Space S : A nonempty countable set. An element is called an outcome. A subset of S is called an event to which a probability is assigned. If you look closely, you will realize that to calculate this probability we first have to count the elements in these sets.

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Probability Counting Rules of counting the elements in a set

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Probability The Addition Rule The basic rule underlying the calculation of the number of elements in a union or difference or intersection is the addition rule. This rule states that the number of elements in a union of mutually disjoint finite sets equals the sum of the number of elements in each of the component sets. Theorem 9.3.1: Suppose a finite set A equals the union of k distinct mutually disjoint subsets A1, A2, …., Ak. Then N(A) = N(A1)+N(A2)+…+ N(Ak)

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Probability The Addition Rule---Cont. Example: A computer access password consists of from one to three letters chosen from the 26 in the alphabet with repetitions allowed. How many different passwords are possible? Solution: The set of all passwords can be partitioned into subsets consisting of those of length 1, those of length 2, and those of length 3 as shown in the figure below.

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Probability The Addition Rule---Cont. By the addition rule, the total number of passwords equals the number of passwords of length 1, plus the number of passwords of length 2, plus the number of passwords of length 3. Now the, Number of passwords of length 1= 26 Number of passwords of length 2 =262

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Probability The Addition Rule---Cont. Number of passwords of length 3 =263 Hence the total number of passwords= 261+262+263=18,278

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Probability The Difference Rule An important consequence of the addition rule is the fact that if the number of elements in a set A and the number in a subset B of A are both known, then the number of elements that are in A and not in B can be computed. Theorem 9.3.2: The Difference Rule: If A is finite set and B is a subset of A, then N(A-B) = N(A) – N(B)

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Probability The Difference Rule---Cont. The difference rule is illustrated below.

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Probability The Difference Rule---Cont. The difference rule holds for the following reason: If B is a subset of A, then the two sets B and A – B have no elements in common and B (A – B) = A. Hence, by the addition rule, N(B) + N(A – B) = N(A). Subtracting N(B) from both sides gives the equation N(A – B) = N(A) – N(B).

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Probability The Difference Rule---Cont. Example: A typical PIN (personal identification number) is a sequence of any four symbols chosen from the 26 letters in the alphabet and the ten digits, with repetition allowed. a. How many PINs contain repeated symbols? b. If all PINs are equally likely, what is the probability that a randomly chosen PIN contains a repeated symbol?

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Probability The Difference Rule---Cont. a. How many PINs contain repeated symbols? Let’s use the board to intuitively explain why the Difference Rule will work here!

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Probability The Difference Rule---Cont. Example --- Cont.: There are 364 = 1,679,616 PINs when repetition is allowed, and there are 36  35  34  33...

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Probability The Difference Rule---Cont. Example --- Cont.: There are 364 = 1,679,616 PINs when repetition is allowed, and there are 36  35  34  33 = 1,413,720 PINs when repetition is not allowed. Thus, by the difference rule, there are 1,679,616 – 1,413,720 = 265,896 PINs that contain at least one repeated symbol.

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Probability The Difference Rule---Cont. b. If all PINs are equally likely, what is the probability that a randomly chosen PIN contains a repeated symbol? So, how would you figure this out?

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Probability The Difference Rule---Cont. Example --- Cont.: There are 1,679,616 PINs in all, and by part (a) 265,896 of these contain at least one repeated symbol. Thus, by the equally likely probability formula, the probability that a randomly chosen PIN contains a repeated symbol is

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Probability The Difference Rule---Cont. An alternative solution to Example 3(b) is based on the observation that if S is the set of all PINs and A is the set of all PINs with no repeated symbol, then S – A is the set of all PINs with at least one repeated symbol.

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Probability The Difference Rule---Cont. We know that the probability that a PIN chosen at random contains no repeated symbol is

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Probability The Difference Rule---Cont. We know that the probability that a PIN chosen at random contains no repeated symbol is And hence

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Probability The Difference Rule---Cont. This solution illustrates a more general property of probabilities: that the probability of the complement of an event is obtained by subtracting the probability of the event from the number 1. Formula for the Probability of the Complement of an event! If S is a finite sample space and A is an event in S, then P(Ac) = 1- P(A).

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Probability The Inclusion/Exclusion Rule The addition rule says how many elements are in a union of sets if the sets are mutually disjoint. Now consider the question of how to determine the number of elements in a union of sets when some of the sets overlap. For simplicity, begin by looking at a union of two sets A and B, as shown below.

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Probability The Inclusion/Exclusion Rule--- Cont. To get an accurate count of the elements in , it is necessary to subtract the number of elements that are in both A and B. Because these are the elements in .

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Probability Counting Rules in terms of Probabilities If {E0, E1, ….} is collection of disjoint events, then

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Probability Counting Rules in terms of Probabilities---Cont. Complement Rule: )

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Probability Counting Rules in terms of Probabilities---Cont. ) )

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Further Counting Counting Subsets of a Set: Combinations: Look at these examples: In how many ways, can I select 5 books from my collection of 100 to take on vacation? How many different ways 13-card Bridge hands can be dealt from a 52-card deck? In how many ways, can I select 5 toppings for my pizza if there are 14 available? What is common in all these questions?

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Counting Subsets of a Set: Combinations---Cont. Is read as “n choose k”

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Why Count Subsets of Set? Example: Suppose we select 5 cards at random from a deck of 52 cards. What is the probability that we will end up having a full house? Doing this using the possibility tree will take some effort.