# Chapter 5. Probability

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 ...

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