# 1.1: Sample Spaces, Events, and Outcomes

**At Grade**Created by: Bruce DeWItt

### Learning Objectives

- Determine the sample space for a given event or series of events
- Produce an organized list of outcomes within a sample space

A **sample space** is a list of all the possible outcomes that may occur. What might happen when you flip a coin? You will either get heads or tails. What will happen when you roll a single die? You will either get a 1, 2, 3, 4, 5, or 6. The sample space for flipping a coin is S={heads, tails}. The sample space for rolling a die is S={1,2,3,4,5,6}

On a coin flip, there are two **outcomes**, heads and tails. There are six different outcomes when considering the **event** of rolling a single die.

#### Example 1

Suppose you roll two dice. Build a 6 by 6 grid to show the different outcomes that might happen when you add the two dice together.

a) What is the sample space for the different sums that you might get?

b) What is the event for this situation?

c) Based on your grid, which outcome occurs most often?

#### Solution

a) The sample space is S={2,3,4,5,6,7,8,9,10,11,12}.

b) The event is the rolling of the two dice.

c) Notice that a total of 7 can occur 6 different ways. A total of 7 is the most likely outcome.

#### Example 2

A child orders breakfast at a restaurant. The restaurant has two choices of drinks: milk and orange juice. The restaurant also has three choices of meat: sausage, ham, and bacon. Suppose the child orders one drink and one type of meat.

a) Give the sample space that shows all the different outcomes for what the child might order.

b) How many different outcomes are possible?

#### Solution

a) For the drinks, use M=Milk and O=Orange Juice. For the meat, use S=Sausage, H=Ham, and B=Bacon. The child might order MS, MH, MB, OS, OH, or OB. The sample space is S={MS, MH, MB, OS, OH, OB}. This list can also be generated using a simple grid as shown on the top of the next page.

b) There are six possible outcomes. This can be found simply by counting the number of results within the sample space.

Sometimes, situations can get a bit too complex to simply make a list or build a grid. A **tree diagram** is a visual organizer that is very effective in handling situations with larger numbers of outcomes. We will introduce this concept here, but we will revisit tree diagrams in greater detail in section 1.2.

#### Example 3

A dart player is trying to hit the bulls-eye with each of three darts that he will throw. Each dart will either hit the bulls-eye or miss the bulls-eye. Use a tree diagram to give the sample space for the different outcomes that may occur.

#### Solution

Build the tree diagram shown below to track what might happen.

The sample space is S={HHH, HHM, HMH, HMM, MHH, MHM, MMH, MMM}

### Problem Set 1.1

1) A single coin is flipped two times.

a) Construct the sample space for this situation.

b) How many different outcomes are possible?

2) A single coin is flipped three times.

a) Use a tree diagram to construct the sample space for this situation.

b) How many different outcomes are possible?

3) A single coin is flipped four times.

a) Construct the sample space for this situation.

b) How many different outcomes are possible?

4) Suppose a 4-sided die is rolled one time. What is the sample space for the result of the roll?

5) Suppose two 4-sided dice are rolled and we keep track of the total on the two dice.

a) Draw a four by four grid that demonstrates the different results for the total of the two dice.

b) What is the sample space for the possible totals of the two dice?

6) Suppose a 4-sided die is rolled two times and we keep track of the *product* when the result from the first die is multiplied by the result from the second die.

a) Draw a four by four grid that demonstrates the different results for the product of the two dice.

b) What is the sample space for the possible products of the two dice?

c) How many different outcomes are possible for the product of the two dice?

d) What outcome occurs most often?