Counting and Probability

Counting

Number of Elements in a Set

The number of elements in a set can be determined by systematic methods of counting.

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 {1,2,3,4,5,6}\lbrace 1, 2, 3, 4, 5, 6 \rbrace, and rolling an even number is an event that can be represented as the set {2,4,6}\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 {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\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: {HHT, HTH, THH}\lbrace \text{HHT, HTH, THH}\rbrace.
  • The event of flipping exactly two tails has three outcomes: {HTT, THT, TTH}\lbrace \text{HTT, THT, TTH}\rbrace.
  • The event of flipping at least two tails has four outcomes: {HTT, THT, TTH, TTT}\lbrace \text{HTT, THT, TTH, TTT}\rbrace.

Tree Diagrams

A tree diagram is an organized way of identifying all the possible outcomes in an experiment that involves multiple steps.

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.

In a tree diagram for the experiment of flipping a coin four times, there are two possible outcomes (heads or tails) for the first toss. There are branches from each of those outcomes to represent the possible outcomes for the second toss, and so on. Tracing the top-most branch of the diagram gives the first outcome of the sample space: HHHH.
This tree diagram shows the sample space for flipping a coin four times. Trace the "branches" of the "tree" to identify each possible outcome. The diagram shows that there are 16 outcomes in the sample space.

Multiplication Principle

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.

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 mm ways to do step 1 and nn ways to do step 2, then there are mnm\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.

Step-By-Step Example
Using the Multiplication Principle with Two Parts
Identify the number of possible outcomes when choosing two cards from a standard deck of playing cards without replacing the first one.
Step 1

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.

Step 2
Determine the product of the number of ways to make each choice.
Total cardsRemaining cards=Possible outcomes5251=2,652\begin{aligned}\\\text{Total cards} \cdot \text{Remaining cards}&=\text{Possible outcomes}\\52 \cdot 51 &= 2\text{,}652\end{aligned}
Solution
There are 2,652 possible outcomes from choosing two cards from a standard deck of playing cards without replacement.
Step-By-Step Example
Using the Multiplication Principle with Three Parts
A sandwich is made from one type of bread, one type of protein, and one type of cheese. Identify the number of possible sandwiches when choosing from 8 types of bread, 5 types of protein, and 7 types of cheese.
Step 1

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.
Step 2
Calculate the product of the number of ways to make each choice.
BreadProteinCheese=Possible outcomes857=280\begin{aligned}\text{Bread} \cdot \text{Protein} \cdot \text{Cheese}&=\text{Possible outcomes}\\8 \cdot 5 \cdot 7 &= 280\end{aligned}
Solution
There are 280 possible sandwiches.