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CS2800-Probability_part_a_v.1

# CS2800-Probability_part_a_v.1 - Discrete Math CS 280 Prof...

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1 Discrete Math CS 280 Prof. Bart Selman [email protected] Module Probability --- Part a) Introduction

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2 Terminology Experiment A repeatable procedure that yields one of a given set of outcomes Rolling a die, for example Sample space The set of possible outcomes For a die, that would be values 1 to 6 Event A subset of the sample experiment If you rolled a 4 on the die, the event is the 4
Probability Experiment : We roll a single die, what are the possible outcomes? {1,2,3,4,5,6} The set of possible outcomes is called the sample space. Depends on what we’re going to ask. Often convenient to choose a sample space of equally likely outcomes. {(1,1),(1,2),(1,3),…,(2,1),…,(6,6)} We roll a pair of dice, what is the sample space?

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4 Probability definition: Equally Likely Outcomes The probability of an event occurring (assuming equally likely outcomes) is: Where E an event corresponds to a subset of outcomes. Note: E S. Where S is a finite sample space of equally likely outcomes Note that 0 ≤ |E| ≤ |S| Thus, the probability will always between 0 and 1 An event that will never happen has probability 0 An event that will always happen has probability 1 S E E p = ) (
5 Probability is always a value between 0 and 1 Something with a probability of 0 will never occur Something with a probability of 1 will always occur You cannot have a probability outside this range! Note that when somebody says it has a “100% probability” That means it has a probability of 1

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Dice probability What is the probability of getting a 7 by rolling two dice? There are six combinations that can yield 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) Thus, |E| = 6, |S| = 36, P(E) = 6/36 = 1/6
Probability Which is more likely: Rolling an 8 when 2 dice are rolled? Rolling an 8 when 3 dice are rolled? No clue.

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Probability What is the probability of a total of 8 when 2 dice are rolled? What is the size of the sample space? 36 How many rolls satisfy our property of interest? 5 So the probability is 5/36 ≈ 0.139.
Probability What is the probability of a total of 8 when 3 dice are rolled? What is the size of the sample space? 216 How many rolls satisfy our condition of interest? C(7,2) So the probability is 21/216 ≈ 0.097.

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10 The game of poker You are given 5 cards (this is 5-card stud poker) The goal is to obtain the best hand you can The possible poker hands are (in increasing order): No pair One pair (two cards of the same face) Two pair (two sets of two cards of the same face) Three of a kind (three cards of the same face) Straight (all five cards sequentially – ace is either high or low) Flush (all five cards of the same suit) Full house (a three of a kind of one face and a pair of another face) Four of a kind (four cards of the same face) Straight flush (both a straight and a flush) Royal flush (a straight flush that is 10, J, K, Q, A)
11 Poker probability: royal flush What is the chance of getting a royal flush?

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