In Chapter 2, I said that a probability is a frequency expressed as a fraction of the sample size. That’s one definition of probability, but it’s not the only one. In fact, the meaning of probability is a topic of some controversy.
We’ll start with the uncontroversial parts and work our way up. There is general agreement that a probability is a real value between 0 and 1 that is intended to be a quantitative measure corresponding to the qualitative notion that some things are more likely than others.
The “things” we assign probabilities to are called events. If E represents an event, then P(E) represents the probability that E will occur. A situation where E might or might not happen is called a trial.
As an example, suppose you have a standard six-sided die and want to know the probability of rolling a six. Each roll is a trial. Each time a six appears is considered a success; other trials are considered failures. These terms are used even in scenarios where “success” is bad and “failure” is good.
If we have a finite sample of n trials and we observe s successes, the probability of success is s/n. If the set of trials is infinite, defining probabilities is a little trickier, but most people are willing to accept probabilistic claims about a hypothetical series of identical trials, like tossing a coin or rolling a die.
We start to run into trouble when we talk about probabilities of unique events. For example, we might like to know the probability that a candidate ...