6.
What is probability?
• Mathematical machinery to deal with
uncertain events
• What does uncertain mean?
• What is an event?
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.
Sample space
A set containing all possible outcomes
from an experiment. Often called S.
An event is a subset of the sample space
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.
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.
Contents
• Numeric (quantitative)
• Non-numeric (qualitative)
• Will need to put both on a common
framework (next week)
12.
Cardinality
• Small (< 10)
• Large, but ﬁnite
• Countably inﬁnite
• Uncountably inﬁnite
• We will follow this order as we develop
increasingly complex mathematical
tools
13.
Events
• An event is a subset of the sample
space
• Set of all possible events is the
power set of S
• Examples
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
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.
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.
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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