TABLE OF CONTENTS In Section 1.1, we suggested sample spaces to model the results of various uncertain measurements. We then said that events are subsets of the sample space. In this section, we add probability to sample space models of some simple systems and compute probabilities of various events.
The goal of probability theory is to provide mathematical machinery to analyze complicated problems in which answers are not obvious. However, for any such theory to be accepted, it should provide answers to simple problems that agree with our intuition. In this section we consider several simple problems for which intuitive answers are apparent, but we solve them using the machinery of probability.
Consider the experiment of tossing a fair die and measuring, i.e., noting, the face turned up. Our intuition tells us that the probability of the ith face turning up is 1/6, and that the probability of a face with an even number of dots turning up is 1/2.
Here is a mathematical model for this experiment and measurement. Let the sample space ? be any set containing six points. Each sample point or outcome ? ? ? corresponds to, or models, a possible result of the experiment. For simplicity, let
Now define the events
and
The event F i corresponds to, or models, the die s turning up showing the ith face. Similarly, the event E models the die s showing a face with an even number of dots. Next, for every subset
