Key takeaway: A sample space is the complete list of every possible outcome of a chance activity, with no outcome missing and none listed twice.
Grade 77.SP.C.8bProbability and Counting
Definition of Sample Space
The sample space of an experiment is the master list of what can happen. Before computing any chance, you first ask: what are all the results? For a coin flip the list is heads and tails. For a die it is the numbers 1 through 6.
When an activity has two or more stages, the list grows fast, so people organize it in a table, a tree or an ordered list of pairs. A good list is complete and has no duplicates. Once it is written down, each event is just a part of it, and the chance of an event comes from counting that part compared with the whole list.
Formula and notation
The set is written with braces, and the number of outcomes is counted with bars:
\[S = \{H, T\}, \qquad |S| = 2\]
For two stages with \(m\) and \(n\) results each, the combined list has \(m \times n\) outcomes.
Key facts
- Every outcome appears once, and together they cover all possibilities.
- Ordered pairs such as (3, T) show the result of each stage in sequence.
- A two-stage activity has the product of the stage counts as its outcomes.
- An event is any chosen part of the list.
Sample Space example
Marco rolls a fair 4-sided die and flips a coin. List the sample space, then find the probability of an even number together with tails.
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Pair each die result with each coin result: 1H, 2H, 3H, 4H, 1T, 2T, 3T, 4T. That is \(4 \times 2 = 8\) outcomes.
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An even number with tails matches 2T and 4T, which is 2 outcomes.
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Probability: \(\frac{2}{8} = \frac{1}{4}\).
There are 8 outcomes, and the probability of an even number with tails is \(\frac{1}{4}\).
Common mistake
Watch out
Asked to list the results of flipping two coins, Rhea writes {HH, HT, TT}, saying HT and TH are the same.
Why it is wrong
When the coins are different, or flipped one after the other, “head then tail” and “tail then head” are separate results. The full list is {HH, HT, TH, TT}, with 4 outcomes. Leaving one out makes the chance of exactly one head look like \(\frac{1}{3}\) when it is really \(\frac{2}{4}\).
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Probability and Counting