Hadley Wickham

Stat310Probability and Statistics

1. Two important facts

2. Syllabus

3. Introduction to probability

4. Definitions & properties

5. Probability as a set function

HadleyHELLO

my name is

had.co.nz/stat310

Introduction to probability

What is probability?

• Mathematical machinery to deal with uncertain events

• What does uncertain mean?

• What is an event?

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!)

Sample space

A set containing all possible outcomes from an experiment. Often called S.

An event is a subset of the sample space

Random experiments• The sequence of dice

rolls until you get a six

• The weather tomorrow

• The next hand in a poker game

• Your final grade in this class

• The next President of the United States

• The length of time until your next sneeze

• My age

• The result of a coin flip

• The weight of a bag of m&m’s

• The sex of a randomly selected member of class

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)

Contents

• Numeric (quantitative)

• Non-numeric (qualitative)

• Will need to put both on a common framework (next week)

Cardinality

• Small (< 10)

• Large, but finite

• Countably infinite

• Uncountably infinite

• We will follow this order as we develop increasingly complex mathematical tools

Events

• An event is a subset of the sample space

• Set of all possible events is the power set of S

• Examples

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

Terminology

• Mutually exclusive

• Exhaustive

• Mutually exclude + exhaustive = partition

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

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?

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