 
Summary: Probabilities and Random Variables
This is an elementary overview of the basic concepts of probability theory.
1 The Probability Space
The purpose of probability theory is to model random experiments so that we can draw inferences
about them. The fundamental mathematical object is a triple (, F, P) called the probability
space. A probability space is needed for each experiment or collection of experiments that we wish
to describe mathematically. The ingredients of a probability space are a sample space , a collection
F of events, and a probability measure P. Let us examine each of these in turn.
1.1 The sample space
This is the set of all the possible outcomes of the experiment. Elements of are called sample
points and typically denoted by . These examples should clarify its meaning:
Example 1. If the experiment is a roll of a sixsided die, then the natural sample space is =
{1, 2, 3, 4, 5, 6}. Each sample point is a natural number between 1 and 6.
Example 2. Suppose the experiment consists of tossing a coin three times. Let us write 0 for heads
and 1 for tails. The sample space must contain all the possible outcomes of the 3 successive tosses,
in other words, all triples of 0's and 1's:
= {0, 1}3
= {0, 1} × {0, 1} × {0, 1}
= {(x1, x2, x3) : xi {0, 1} for i = 1, 2, 3}
= {(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}.
