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# Introduction

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### Introduction

1. 1. Stat310 Probability and Statistics Hadley Wickham
2. 2. 1. Two important facts 2. Syllabus 3. Introduction to probability 4. Deﬁnitions & properties 5. Probability as a set function
3. 3. HE LLO my name is Hadley
5. 5. Introduction to probability
6. 6. What is probability? • Mathematical machinery to deal with uncertain events • What does uncertain mean? • What is an event?
7. 7. Random experiment An observation that is uncertain: we don’t know ahead of time what the answer will be (pretty common!) Ideally we want the experiment to be repeatable under exactly the same initial conditions (pretty rare!)
8. 8. Sample space A set containing all possible outcomes from an experiment. Often called S. An event is a subset of the sample space
9. 9. Random experiments • The sequence of dice • The length of time until rolls until you get a six your next sneeze • The weather tomorrow • My age • The next hand in a • The result of a coin ﬂip poker game • The weight of a bag of • Your ﬁnal grade in this m&m’s class • The sex of a randomly • The next President of selected member of the United States class
10. 10. Your turn • How could you classify these different experiments based on the sample space? • Think (2 min) • Pair (3 min) • Square (3 min) • Share (2 min)
11. 11. Contents • Numeric (quantitative) • Non-numeric (qualitative) • Will need to put both on a common framework (next week)
12. 12. Cardinality • Small (< 10) • Large, but ﬁnite • Countably inﬁnite • Uncountably inﬁnite • We will follow this order as we develop increasingly complex mathematical tools
13. 13. Events • An event is a subset of the sample space • Set of all possible events is the power set of S • Examples
14. 14. Set algebra • Intersection and union are: • Commutative (order from left to right doesn’t matter) • Associative (order of operation doesn’t matter) • Distributive (can expand brackets) • You should be familiar with everything on: http://en.wikipedia.org/wiki/Algebra_of_sets
15. 15. Terminology • Mutually exclusive • Exhaustive • Mutually exclude + exhaustive = partition
16. 16. How do we deﬁne uncertainty? • Associate a probability with each element of the sample space. • Deﬁned by the function probability mass function (pmf). • The probability is the long run relative frequency
17. 17. Properties of pmf • What are some properties that the pmf must have? (Use your common sense) • For example, take the random experiment of ﬂipping two coins and observing whether they come up heads or tails. How are the probabilities of the different events related?
18. 18. Properties of pmf • Basic (as deﬁned by book) • Important derived properties (T 1.2-1 - T1.2-6) • Strategies of T1.2-3 and T1.2-5 particularly important
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