Sample space

Here you will learn about sample space, including what it is and how to use it to find probabilities.

Students will first learn about sample space as part of 7 th grade in statistics and probability and continue to learn about it in high school.

What is a sample space?

A sample space is a list, diagram or table of values used to display the set of all possible outcomes.

To create a sample space diagram, think about the possible outcomes of a situation.

For example,

Let’s say you were to flip a fair coin with outcomes H (heads) or T (tails) and at the same time roll a fair six-sided die with outcomes: 1, 2, 3, 4, 5, 6.

The sample space for this situation could be written as a list of the possible combinations: H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6.

The sample space for more complicated situations is often shown in a table.

For example,

If flipping the coin and rolling the die were part of a game and the side the coin shows changed the score of the die, you could show this in a sample space diagram. In this game, the coin showing heads adds 1 to the value on the die. The coin showing tails doubles the value on the die.

The sample space diagram shows all the possible outcomes for this game.

You can use sample space diagrams to find probabilities.

Jon plays the game and wants to know the probability of getting a score of more than 6.

You can see from the table that there are 4 scores more than 6.

There are a total of 12 possible outcomes from this game.

Using the theoretical probability formula,

P( score more than 6) =\cfrac{4}{12}=\cfrac{1}{3}.

What is a sample space?

Common Core State Standards

How does this relate to 7 th grade math?

• Grade 7: Statistics and Probability (7.SP.C.8)
  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
  • a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
  • b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (example, “rolling double sixes”), identify the outcomes in the sample space which compose the event.
  • Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40\% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

How to create a sample space

In order to create a sample space:

1. Use information provided to decide whether to write a list or create a table to find all possible outcomes.
2. Systematically write the list or fill in the table by either listing outcomes or performing operations with values.

Sample space examples

Example 1: sample space of a simple event

A card is drawn from a standard deck of 52 playing cards. Show a sample space for all possible outcomes.

1. Use information provided to decide whether to write a list or create a table to find all possible outcomes.

Playing cards have four different suites (clovers or clubs ♣, diamonds ♦, hearts ♥, and spades ♠. Each suite has numbers 2–10 and J, Q, K and A. As there are a large number of cards with some variety, a table can be used to organize the sample space.

2Systematically write the list or fill in the table by either listing outcomes or performing operations with values.

One way to systematically fill in the table is to make each suite a row. Then you can just list the same card numbers/letters for each suite.

Example 2: sample space of a compound event

Show the sample space for all possible outcomes of flipping a coin and spinning the spinner.

Example 3: sample space of a compound event

Show the sample space for all possible outcomes of rolling a die twice.

Systematically write the list or fill in the table by either listing outcomes or performing operations with values.

To systemically fill in the table, make the first row the outcomes for the first roll and the first column the outcome for the second roll. Also list the options 1–6 in order.

The sample space is all the cells that show the possible outcomes of two rows (outlined in blue). For example, if the first roll is a 2 (right arrow) and the second roll is a 3 (down arrow), the outcome can be found where the two arrows meet.

How to use a sample space

In order to find a probability using a sample space diagram:

1. Use information provided to decide whether to write a list or create a table to find all possible outcomes.
2. Systematically write the list or fill in the table by either listing outcomes or performing operations with values.
3. Use the information from the list or table to find any probabilities required.

Example 4: writing a list of possible outcomes

A fair coin is flipped and the spinner is spun.

Write a list of all possible combinations of outcomes and find the probability of getting a tail and an even number.

Example 5: creating a sample space table

Two fair spinners A and B, one numbered 1, 2 and 3, the other 2, 5, and 7 are spun.

The scores on the spinners are added together. Find the probability that the total score is a prime number.

Use information provided to decide whether to write a list or create a table to find all possible outcomes.

Each spinner has 3 outcomes. You can use a table to show all combinations of the spinners.

Systematically write the list or fill in the table by either listing outcomes or performing operations with values.

Use the information from the list or table to find any probabilities required.

There are three possible outcomes that are prime.

The probability is the ratio of the desired subset (prime) over the total possible outcomes.

P( total score is prime ) =\cfrac{3}{9}

Example 6: creating a sample space diagram using a rule

A fair coin is flipped and a fair six-sided die is rolled. If the coin shows heads, the score on the die is increased by 3. If the coin shows tails the score on the die is decreased by 1. Find the probability of getting a score less than 5.

Use information provided to decide whether to write a list or create a table to find all possible outcomes.

This question is best answered with a table, so you can keep track of the combination of flips/rolls and the total points.

Systematically write the list or fill in the table by either listing outcomes or performing operations with values.

Use the information from the list or table to find any probabilities required.

There are six possible outcomes that are less than 5.

The probability is the ratio of the desired subset ( less than 5) over the total possible outcomes.

P( score less than 5)=\cfrac{6}{12}=\cfrac{1}{2}

Teaching tips for sample space

• Introduce sample space with simple and compound probabilities. Use small, finite sample spaces, such as rolling a die and getting an odd number (simple) or tossing a coin and flipping a coin (compound). Creating sample spaces is the main way that students solve compound probabilities without the rule.

• Help students develop their statistical inference skills by having them estimate the total number of outcomes or the probability of the event before actually creating a sample space. Once they have solved, have them revisit these estimates and discuss how accurate they were and why.

• Once students are comfortable with creating sample spaces, have them complete random experiments to see how the experimental probability compares to the theoretical probability.
  
  Discuss with students why the outcome of the experiment will not always match the calculated probability. This can help them grow ideas about probability space, probability theory and other theorems that will be fostered in more advanced probability topics.

Easy mistakes to make

• Missing out or repeating outcomes when listing
  A common error is to miss out some outcomes or repeat one already listed. This leads to the incorrect number of outcomes. It can happen if you do not use a systematic approach to forming the list. Creating a table to list the outcomes can help structure our list to ensure none are missed or repeated.

• Confusing the order of the ratio when calculating the probability of an event
  To calculate the probability, write the ratio with the desired outcomes as the numerator and all possible outcomes as the denominator. Confusing this order will lead to the incorrect probability of any event.

Related compound probability lessons

• Fundamental counting principle
• Sample space

The next lessons are

• Probability distribution
• Units of measurement
• Represent and interpret data

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