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.

Sample Space 1 US

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.

Sample Space 2 US

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?

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, β€œrolling 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?

[FREE] Probability Worksheet (Grade 7 to 12)

[FREE] Probability Worksheet (Grade 7 to 12)

[FREE] Probability Worksheet (Grade 7 to 12)

Use this quiz to check your grade 7 to 12 students’ understanding of probability. 15+ questions with answers covering a range of 7th to 12th grade probability topics to identify areas of strength and support!

DOWNLOAD FREE
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[FREE] Probability Worksheet (Grade 7 to 12)

[FREE] Probability Worksheet (Grade 7 to 12)

[FREE] Probability Worksheet (Grade 7 to 12)

Use this quiz to check your grade 7 to 12 students’ understanding of probability. 15+ questions with answers covering a range of 7th to 12th grade probability topics to identify areas of strength and support!

DOWNLOAD FREE

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.

Sample Space 3 US

Playing cards have four different suites (clovers or clubs ♣, diamonds ♦, hearts β™₯, and spades β™ . 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.

Sample Space 4 US

Example 2: sample space of a compound event

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

Sample Space 5 US

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

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

Example 3: sample space of a compound event

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

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

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

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.

Sample Space 8 US

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

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

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.

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.

Sample Space 9 US

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.

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.

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.

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.

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.

Practice sample space questions

1. Ellie has a bag with 1 green marble, 2 red marbles and 4 blue marbles.

 

Which choice shows the sample space of randomly picking one marble from Ellie’s bag?

Green, Red, Blue

GCSE Quiz False

Green, Red -1, Red -2, Blue -1, Blue -2, Blue -3, Blue -4

GCSE Quiz True

Green, Red, Blue, Blue

GCSE Quiz False

Green -1, Red -2, Blue -4

GCSE Quiz False

If you were to look into Ellie’s bag, this is what you would see.

 

Sample Space 16 US

 

The sample space needs to show that there is more than one red and blue marble.

 

You can number each red and blue to show that there is more than one.

 

Sample Space 17 US

2. A fair coin is flipped and the spinner is spun.

 

Sample Space 18 US

 

How many possible outcomes are there?

2
GCSE Quiz False

5
GCSE Quiz False

6
GCSE Quiz True

7
GCSE Quiz False

You can create a list showing all possible outcomes. Show the outcomes for heads (H), then tails (T).

 

H-A, H-B, H-C

 

T-A, T-B, T-C

3. A fair coin is flipped and the spinner shown below is spun. If the coin shows heads the score on the spinner is doubled. If the coin shows tails the score on the spinner is tripled. Which shows the sample space for this situation?

 

Sample Space 19 US

Sample Space 20 US

GCSE Quiz True

Sample Space 21 US

GCSE Quiz False

Sample Space 22 US

GCSE Quiz False

Sample Space 23 US

GCSE Quiz False

A table works best, since you are applying an operation to each score on the spinner.

 

The expanded version of the table below, shows how the sample space was calculated.

 

Sample Space 24 US

4. The sample space diagram shows the possible outcomes when two normal fair dice are rolled and the difference between values is calculated.

 

Sample Space 25 US

 

Use the sample space diagram to find the probability of getting a difference of 2 or less.

\cfrac{5}{12}
GCSE Quiz False

\cfrac{1}{3}
GCSE Quiz False

\cfrac{2}{9}
GCSE Quiz False

\cfrac{2}{3}
GCSE Quiz True

There are 24 outcomes that are 2 or less out of a total of 36.

 

\cfrac{24}{36} = \cfrac{2}{3}

5. Three fair coins are flipped together. Each coin has either a head or a tail. Find the probability of getting 2 heads and one tail.

\cfrac{2}{3}
GCSE Quiz False

\cfrac{3}{8}
GCSE Quiz True

\cfrac{1}{4}
GCSE Quiz False

\cfrac{3}{4}
GCSE Quiz False

Below is the sample space of three coins flipped:

 

Sample Space 26 US

 

There are 3 outcomes which have 2 heads and 1 tail, out of 8 outcomes in total.

6. Two bags each contain 3 numbered counters. Bag A contains the numbers 2, 7 and 9. Bag B contains the numbers 1, 3 and 5.

 

A counter from each bag is selected at random and their values multiplied together. Find the probability of getting a result greater than 10.

\cfrac{4}{9}
GCSE Quiz True

\cfrac{5}{9}
GCSE Quiz False

\cfrac{1}{2}
GCSE Quiz False

\cfrac{2}{3}
GCSE Quiz False

Below is the sample space of choosing a numbered counter from each bag and finding their product:

 

Sample Space 27 US

 

There are 4 outcomes greater than 10, out of 9 outcomes in total.

Sample space FAQs

What is the intersection of two independent events?

The intersection of two independent events is when both the events happen together. This can be modeled with a venn diagram or calculated using the rule.

See also: Independent events

What is conditional probability?

The probability of a second event, given that the first event has already occurred.

See also: Conditional probability

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