### Number of Elements in a Set

An **experiment** is an activity involving chance in which the outcomes are used to estimate probability. When performing an experiment, like rolling a die or flipping a coin, there is a set of possible outcomes. A **set** is a collection of objects, and an **outcome** is a possible result of an experiment. The **sample space** for an experiment is the set of all possible outcomes. An **event** is a subset of the outcomes in the sample space of an experiment. For the experiment of rolling a die, the sample space is $\lbrace 1, 2, 3, 4, 5, 6 \rbrace$, and rolling an even number is an event that can be represented as the set $\lbrace 2,4, 6\rbrace$.

Making an organized list is one way to determine the number of elements in a set. When a coin is flipped three times, there are a variety of possible outcomes. To list the outcomes in the sample space, let H represent the outcome for heads and let T represent the outcome for tails. The sample space is $\lbrace \text{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\rbrace$.

The list is organized by recording all outcomes with zero Ts, then all outcomes with one T, then all outcomes with two Ts, and then all outcomes with three Ts. There are eight possible outcomes in the sample space.

- The event of flipping exactly one tail has three outcomes: $\lbrace \text{HHT, HTH, THH}\rbrace$.
- The event of flipping exactly two tails has three outcomes: $\lbrace \text{HTT, THT, TTH}\rbrace$.
- The event of flipping at least two tails has four outcomes: $\lbrace \text{HTT, THT, TTH, TTT}\rbrace$.

### Tree Diagrams

A **tree diagram** displays the sample space of an experiment so that the possible outcomes can be determined and counted. To make a tree diagram, draw a branch to represent the possible outcomes in each stage of the experiment. Trace each branch to the end of the diagram to determine each outcome in the sample space.

### Multiplication Principle

The multiplication principle, or fundamental counting principle, is a rule for counting the number of outcomes in an experiment. A probability experiment can be thought of as a process. The **multiplication principle** states that for a process involving two steps, if there are $m$ ways to do step 1 and $n$ ways to do step 2, then there are $m\cdot n$ ways to complete the process. For an experiment, the number of possible outcomes is the number of ways the process can be completed.

When there are more than two steps, the multiplication principle still applies; multiply the number of ways each step can be done to find the total number of possible ways to complete the process.

Calculate the number of ways to choose each of the two cards. There are 52 cards in a standard deck of playing cards.

First, choose one card from 52 cards. Then, choose a second card from the remaining 51 cards.

There are three different choices to make:

- Choose from eight types of bread.
- Choose from five types of protein.
- Choose from seven types of cheese.