### Probability Definitions

**Probability** is the likelihood that an event will happen. The probability of an event is always between zero and one, or between 0% and 100%. An event with a probability of zero is impossible, and an event with a probability of one is certain. Events with a probability closer to zero are unlikely, while those with a probability closer to one are likely. Events with the same probability are equally likely.

The probability of an event can be written as $P(\text{event})$, which is read as "the probability of the event." In an experiment that involves flipping a coin, the notation $P(\text{heads})$ or $P(H)$ may be used to denote the probability that the outcome is heads.

The probabilities of all the outcomes in the sample space of an event have a sum of 1 because it is certain that one of the outcomes will happen. The **complement of an event** is all outcomes in the sample space that are not the event. For example, if an event $(E)$ is rolling an even number on a die, then the complement of that event $(\text{not } E)$ is not rolling an even number, which is the same as rolling an odd number.

### Experimental Probability

**Experimental probability**is an estimate of the likelihood of an event that is based on observations in an experiment. Each repetition of the experiment is called a trial. When many trials are performed, the experimental probability of an event is the ratio of the number of times the event occurs to the number of trials:

### Theoretical Probability

**Theoretical probability**is a likelihood of an event that is determined by using a model. One common model is to assume that all outcomes are equally likely. Then the theoretical probability of an event is the ratio of the number of ways the event can occur to the total number of equally likely outcomes:

Result of Roll | Frequency | Experimental Probability |
---|---|---|

$1$ | $12$ | $\frac{12}{50}=\frac{6}{25}$ |

$2$ | $7$ | $\frac{7}{50}$ |

$3$ | $9$ | $\frac{9}{50}$ |

$4$ | $5$ | $\frac{5}{50}=\frac{1}{10}$ |

$5$ | $9$ | $\frac{9}{50}$ |

$6$ | $8$ | $\frac{8}{50}=\frac{4}{25}$ |

Result of Spin | Frequency |
---|---|

Red | 8 |

Blue | 18 |

Yellow | 14 |

Result of Spin | Frequency | Experimental Probability |
---|---|---|

Red | $8$ | $\frac{8}{40}$ |

Blue | $18$ | $\frac{18}{40}$ |

Yellow | $14$ | $\frac{14}{40}$ |

Result of Spin | Theoretical Probability |
---|---|

Red | $\frac{1}{5}$ |

Blue | $\frac{2}{5}$ |

Yellow | $\frac{2}{5}$ |

Result of Spin | Frequency | Experimental Probability | Theoretical Probability |
---|---|---|---|

Red | $8$ | $\frac{8}{40}=20\%$ | $\frac{1}{5}=20\%$ |

Blue | $18$ | $\frac{18}{40}=45\%$ | $\frac{2}{5}=40\%$ |

Yellow | $14$ | $\frac{14}{40}=35\%$ | $\frac{2}{5}=40\%$ |