# Chapter 2: Conditional Probability

**Basic**Created by: CK-12

## Introduction

This chapter builds on the concepts learned in the previous chapter on probability. Starting with tree diagrams as a means of displaying outcomes for various trials, we will learn how to read the diagrams and find probabilities. We will also find that order does not matter unless working with permutations. Permutations, such as the combination of your lock at the gym, require their own special formula. When outcomes for permutations have repetitions, these repetitions must also be included in the calculations in order to account for the multiple entries. Combinations, like permutations, also have their own special formula. Combinations, such as the number of teams of 4 that can be arranged in a class of 15 students, are different from permutations, because the order in combinations is insignificant. In this chapter, we will also learn about conditional probability. Conditional probability comes into play when the probability of the second event occurring is dependent on the probability of the first event.

## Chapter Outline

- 2.1. Tree Diagrams
- 2.2. Permutations and Combinations Compared
- 2.3. Permutation Problems
- 2.4. Permutations with Repetition
- 2.5. Combinations
- 2.6. Combination Problems
- 2.7. Conditional Probability

### Chapter Summary

## Summary

This chapter begins with learning how to draw and use tree diagrams as a way to visualize and calculate probabilities. It then moves on to the more complex caculations of permuations and combinations - where permutations are events where the order matters, and combinations are events where the order does *not* matter. Given \begin{align*}n\end{align*} total number of objects, and \begin{align*}r\end{align*} objects chosen is expressed as \begin{align*}{_n}P_r\end{align*}, the number of *permutations* (with order) is:

\begin{align*}{_n}P_r=\frac{n!}{(n-r)!}\end{align*}

while the number of *combinations* (without order) is:

\begin{align*}{_n}C_r = \frac{n!}{r!(n-r)!}\end{align*}

The chapter concludes with how to calculate using conditional probability, where a second event depends on the first event.