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Introduction to Probability: Counting Methods Rutgers University Discrete Mathematics for ECE 14:332:202 Why Probability? We can describe processes for which the outcome is uncertain By their average behavior By the likelihood of particular outcomes Allows us to build models for many physical behaviors Speech, images, traffic … Applications Communications Speech and Image Processing Machine Learning Decision Making Network Systems Artificial Intelligence Used in many undergraduate courses (every grad course) Methods of Counting One way of interpreting probability is by the ratio of favorable to total outcomes Means we need to be able to count both the desired and the total outcomes For illustration, we explore only the most important applications: Coin flipping Dice rolling Card Games Combinatorics Mathematical tools to help us count: How many ways can 12 distinct objects be arranged? How many different sets of 4 objects be chosen from a group of 20 objects? -- Extend this to find probabilities … Combinatorics Number of ways to arrange n distinct objects n! Number of ways to obtain an ordered sequence of k objects from a set of n: n!/(n-k)! -- k permutation Number of ways to choose k objects out of n distinguishable objects: n n! k k!(n k )! This one comes up a lot! Set Theory and Probability We use the same ideas from set theory in our study of probability Experiment Outcome – any possible observation of an exp. Roll a six Sample Space – the set of all possible outcomes Roll a dice 1,2,…6 Event – set of outcomes Dice rolled is odd Venn Diagrams Outcomes are mutually exclusive – disjoint S 2 1 4 3 5 6 Event A Outcomes An Example from Card Games What is the probability of drawing two of the same card in a row in a shuffled deck of cards? Experiment Event Space Pulling two cards from the deck All outcomes that describe our event: Two cards are the same Sample Space All Possible Outcomes All combinations of 2 cards from a deck of 52 Sample Space/Event Space Venn Diagram Event Space (set of favorable outcomes) S all possible outcomes {A,A} {K,2} Calculating the Probability # outcomes _ in _ event _ space P(Event) = # outcomes _ in _ sample _ space Expressed as the ratio of favorable outcomes to total outcomes -- Only when all outcomes are EQUALLY LIKELY Probabilities from Combinations Rule of Product: 52 52! 1326 2 2 ! ( 52 2 )! Total number of two card combinations? We need to find all the combinations of suit and value that describe our event set: use rule of product to find the number of combinations First, we find number of values – 13 choices, and choices of suits: 4 4! 6 2 2!(4 2)! to give our number of possible outcomes 13*6 = 78 Probability(Event) = 78/1326 = 0.0588 Probabilities from Subexperiments Only holds for independent experiments Let’s look at the last problem: Two subexperiments: First can be anything 52/52 = 1 Second, must be one of the 3 remaining cards of the same value from 51 remaining cards 3/51 = 0.588 An Example from Dice Rolling Experiment: Roll Two (6-sided) Dice Event: Numbers add to 7 Sample Space: (all possible outcomes) S = 1,1 1, 2 1,3 1, 4 1,5 1, 6 2,1 3,1 4,1 5,1 6,1 2, 2 3, 2 4, 2 5, 2 6, 2 2,3 3,3 4,3 5,3 6,3 2, 4 3, 4 4, 4 5, 4 6, 4 2,5 3,5 4,5 5,5 6,5 2, 6 3, 6 4, 6 5, 6 6, 6 Sample Space/Event Space Event Space Venn Diagram 1,1 2,1 3,1 4,1 5,1 6,1 1, 2 2, 2 3, 2 4, 2 5, 2 6, 2 1,3 2,3 3,3 4,3 5,3 6,3 1, 4 2, 4 3, 4 4, 4 5, 4 6, 4 1,5 2,5 3,5 4,5 5,5 6,5 1, 6 2, 6 3, 6 4, 6 5, 6 6, 6 S Calculating Probability P(Event) = # outcomes _ in _ event _ space # outcomes _ in _ sample _ space = 6/36 = 1/6 Side Note Probability is something we calculate “theoretically” as a value between 0 and 1, it is not something calculated through experimentation (that is more statistics). Just because you roll a dice 100 times, and it came up as a 1 20 times, does not make P(roll a 1) = 0.2 It would be the limiting case in doing an infinite number of experiments, but this is impossible. So, call your calculated values the “probability”, and your experimental values the “relative frequency”.