# Counting and Probability

## Overview

### Description

Counting and probability are used to determine how likely something is to happen. There are a variety of ways to count the number of ways something can happen, including diagrams and formulas. Counting is needed to determine probabilities. The probabilities of events are figured differently depending on the situation. The theoretical probability is what is expected to happen, but it doesn't always match the experimental probability, or what actually happens.

### At A Glance

• The number of elements in a set can be determined by systematic methods of counting.
• A tree diagram is an organized way of identifying all the possible outcomes in an experiment that involves multiple steps.
• In a process with multiple steps, the total number of ways to complete the process is the product of the number of ways each step can be done.
• A factorial of a natural number represents a product of all the natural numbers less than or equal to the number.
• A permutation is a selection of elements of a set such that order matters; the number of permutations of $n$ objects chosen $r$ at a time can be determined by a formula.
• A combination is a selection of elements of a set such that order does not matter. The number of combinations of $n$ objects chosen $r$ at a time can be determined by a formula.
• The probability of an event is a number from zero to one (or a percentage from 0% to 100%) that describes how likely the event is to occur.
• Experimental probability is an estimate of how likely an event is to occur. It is based on observations of how frequently the event occurs during an experiment.
• Theoretical probability is a calculation of how likely an event is to occur. It is based on a theoretical model, such as assuming that all outcomes are equally likely.
• A probability model is used to determine the probability of each outcome in a sample space.
• Determining whether an event is independent or dependent relies on whether the probability of one event affects the probability of the other event.
• When two events are mutually exclusive, the probability that either event will occur is the sum of the probabilities of each event.
• When two events are mutually inclusive, the probability that either event will occur is the sum of the probabilities of each event minus the probability that both occur together.
• A variety of methods can be used to calculate probabilities of events.