# Probability Basics

Probability is a measure of the likelihood of some event happening. (Just about anything can be an event in this sense.) We measure on a scale from 0 to 1 (or 0% to 100%), where smaller numbers indicate less likelihood and higher numbers indicate greater likelihood. This system in an example of how mathematics is used to formalize and make precise our informal notions about some things being more or less likely to occur than other things.

## Two kinds of probability

We can distinguish two kinds of probability. Mathematical probability is the measure of the relative frequency of an event occurring. In addition, we will use the term personal probability for a statement of someone's degree of belief that an event will occur.

## Determining Relative Frequency (Mathematical Probability)

### Empirical Method

If the process under study can be repeated or simulated many times, we can determine the empirical probability by keeping track of the outcomes in our (large number of) trials. The probability assigened is

P(A happens) = (# times A happened) / (# trials)

If the number of trials is very large, then it is quite likely that this will give us a reliable estimate.

### Theoretical Method

Sometimes we can make mathemitical assumptions about a situation and use Four Basic Properties of Probability to determine the theoretical probability of an event. The accuracy of a theoretical probability depends on the validity of the mathematical assumptions made.

The four useful rules of probability are:

1. It happens or else it doesn't. The probabilty of an event happening added the probability of it not happing is always 1.

P(A happens) + P(A doen't happen) = 1

2. Exclusivity. If A and B can't both happen at the same time (in which case we say that A and B are mutually exclusive), then

P(either A or B happens) = P(A happens) + P(B happens)

3. Independence. If B is no more or less likely to happen when A happens than when A doesn't (in which case we say that A and B are independent), then

P(A and B both happen) = P(A happens) * P(B happens)

4. Sub-Events. If whenever A happens B must also happen, then B must be at least as likely as A, so

P(A happens) <= P(B happens)

Empirical probabilities will also follow these rules (for a given set of trials). Becuase people often have a poor sense of the likelihood of an event, personal probabilities often do not follow these rules. A collection of personal probabilities is called coherent if it does not violate the rules for mathematical probability.

### Equally likely outcomes

One especially important use of these probability rules is the conclusions that can be drawn if we assume that a number of events are equally likely. If there are only n such events that are possible in a given situation, and all are equally likely and pairwise mutually exclusive (no two can happen at once), then each must have probability 1/n. More complicated situations can be handled by dividing a situation into a number of equally likely outcomes and counting how many of them are "of interest" (in the event). The probabilty then is given by (number of interest)/(total number), just as in the case for empirical probability.