Sample Space
In short: Sample space is the complete list of every outcome that can happen in a chance experiment, and counting it gives the bottom number of a probability fraction.
7th GradeProbability7.SP.C.8b
What is Sample Space?
The sample space is the full inventory of possibilities that must be laid out before anyone can say how likely something is: each outcome appears exactly once, and nothing that could occur is missing.
For one coin toss it reads \(\{H, T\}\). Pair a coin with a four-sided die and every pairing counts, giving \(2 \times 4 = 8\) outcomes. When all outcomes are equally likely, an event’s chance equals how many outcomes it picks out, over how many the whole inventory holds.
How to spot it
- A problem says list all the possible outcomes, or asks how many different results can occur.
- Two or more actions happen together, like a toss and a spin, and every pairing must be counted.
- You are asked for the probability of an event and must first know how many outcomes there are in all.
- A table, organized list or diagram is requested to avoid missing cases.
The method
- Name each stage of the experiment and the choices available at that stage.
- Organize the pairings in a table or an ordered list so none is skipped or repeated.
- Count the entries; when stages are independent, multiplying the number of choices gives the same total.
- Circle the entries that belong to your event and form the fraction circled over total.
Sample Space: solved example
Problem
Mia flips a quarter, then rolls a pyramid-shaped die with faces 1 to 4. Count the entries of the full inventory and find the chance of heads together with an even roll.
Solution
Two choices for the coin and four for the die give \(2 \times 4 = 8\) equally likely pairings, as the table in the figure shows.
Heads with an even roll fits only H2 and H4, so 2 pairings. The chance is \(\frac{2}{8} = \frac{1}{4}\), and the inventory holds 8 outcomes.
Try it
Question 1
A carnival game uses one spinner with 3 equal sections and another with 5 equal sections. How many outcomes does the sample space contain?
Show the answer
\(3 \times 5 = 15\) outcomes.
Question 2
Two different letters are picked in order from M, A, T, H. How many ordered outcomes exist?
Show the answer
4 choices for the first letter and 3 for the second: \(4 \times 3 = 12\).
Sample Space vs. Compound Event
The sample space is the whole list of possible results; it is the universe you draw from. A compound event is a particular combination of results inside that universe, such as heads and an even number together. One is the set of everything, the other a chosen subset.
Keep going
Lessons
Worksheets and quizzes
- Compound Events and Simulations – Worksheet
- Compound Events and Simulations – Quiz
- Probability and Probability Models – Worksheet
- Probability and Probability Models – Quiz
Related words
Whole dictionary: A–Z · 7th Grade · Probability