MATH 180A Lecture 2

Uniform Probability Measure

Let be a set with . The uniform probability measure on is defined by
= cardinality of = number of elements in
A probability is uniform for every .

TODO: Prove this equivalence using the three axioms of probability.

Example. A fair die is rolled twice.

a) Describe the sample space and probability measure

: Cartesian product, means all ordered pairs
= uniform probability measure on .

b) Calculate the probability that the sum of two rolls is 8.

c) Calculate the probability that the 2nd roll is strictly larger than the 1st roll.

Key Property of Probability Space: If is a (finite or infinite) sequence of disjoint events, then

Example. Suppose 2 fair coins are flipped. What is the probability that both are heads or both are tails?

Suppose we flip 2 fair coins again. Let = “at least one heads” =
and = “at least one tails” =

Then
The formula doesn’t hold because
not disjoint

Inclusion-Exclusion Principle (2 events)

For any 2 events in any probability space,

Proof.

Section 1.2

Random Sampling

We successively select elements from a finite set () and record the outcome.

Example:

Variant 1: w/replacement, order matters

Variant 2: w/o replacement, order matters

= set of distinct -tuples from .
” permute “

Variant 3: w/o replacement, order doesn’t matter (use to show that order doesn’t matter)

= “set of unordered distinct -tuples from ” = “set of subsets of w/ cardinality ”
” choose “

The probabilitiy measure will consider for these problems is uniform