If the proportion of occurrences of an outcome settles down to one value over the long run, that one value is then defined to be the **probability** of that outcome. Probabilities can be expressed as fractions (5/8), decimals (0.625), or percents (62.5%). There are two main rules that probabilities must satisfy for a given experiment:

- The probability of any event must be greater than or equal to 0 and less than or equal to 1. In symbols:
**0 ≤ P ≤ 1**. For example, it does not make sense to say that there is a “**–30%**” chance of rain, nor does it make sense to say that there is a “**140%**” chance of rain. - The sum of the probabilities of all possible outcomes must equal 1. In other words, if we examine all possible outcomes from an experiment, one of them must occur! It does not make sense to say that there are two possible outcomes, one occurring with probability 20% and the other with probability 50%. What happens the other 30% of the time?

If an event is *impossible*, then its probability must be equal to 0 (i.e. it can never happen). If an event is a *certainty*, then its probability must be equal to 1 (i.e. it always happens).

A **probability model** is a mathematical description of long-run regularity consisting of a sample space ** S** and a way of assigning probabilities to events. Probability models must satisfy both of the above rules. There are two main ways to assign probabilities to outcomes from a sample space:

- The
**empirical method**, in which an experiment is repeated over and over until you have an idea what the probabilities are for each outcome. - The
**classical method**, which relies on*counting*techniques to determine the probability of an event.

##### Example

A basketball player shoots three free throws. We are interested in creating a probability model for the number of free throws that a basketball player makes when shooting three in a row. Recall from above that the sample space for this event is:

*S* = {*HHH, HHM, HMH, HMM, MHH, MHM, MMH, MMM*}

If we count the numbers of hits (H) for each possible outcome, we would get:

*S * = {3, 2, 2, 1, 2, 1, 1, 0}

The probability model for the number of free throws made, assuming this player has an equal chance of making (hitting) or missing the free throw, is:

Hits |
Probability (Fraction) |
Probability (Decimal) |
Probability (Percent) |

0 | 1 out of 8 = 1/8 | 0.125 | 12.5% |

1 | 3 out of 8 = 3/8 | 0.375 | 37.5% |

2 | 3 out of 8 = 3/8 | 0.375 | 37.5% |

3 | 1 out of 8 = 1/8 | 0.125 | 12.5% |