The sample space of a chance experiment is the set of all its possible outcomes, usually written S. For one coin flip, S = {H, T}; for one number cube, S = {1, 2, 3, 4, 5, 6}. Organized lists, tables and tree diagrams help you write it out, and the probabilities of all its outcomes add to 1.
A tree diagram makes sure no outcome is missed: every path from the start to an end is one outcome of the sample space.
Formal definition and notation
The sample space S of a chance process is the set whose elements are all the possible outcomes, listed so that exactly one of them occurs on each trial. Every event is a subset of S, and P(S) = 1. For a multi-step experiment, outcomes are ordered lists of results, and if step 1 has m results and step 2 has n, the sample space has m × n outcomes (the fundamental counting principle).
List every outcome exactly once: no repeats, nothing missing.
Order matters for steps: HT (heads, then tails) and TH are different outcomes.
Three tools: an organized list, a table (good for two steps) and a tree diagram (good for two or more steps).
Probabilities of all outcomes in S add to 1.
If the outcomes are equally likely, P(A) = n(A)n(S) for any event A.
The same experiment can have different sample spaces depending on what you record: the sum of two dice (2 to 12) or the pair of faces (36 pairs).
Counting the sample space
Counting the sample space
n(S) = m × n for two steps with m and n results
m
number of results of the first step
n
number of results of the second step
Keep multiplying for more steps: 3 coin flips give 2 × 2 × 2 = 8 outcomes.
Sample Space examples, step by step
Example 1: List a two-step sample space
Warm-up
A spinner has sections A, B and C. You spin it and flip a coin. Write the sample space.
Pair each letter with H and with T.
S = {AH, AT, BH, BT, CH, CT}.
Check the count: 3 × 2 = 6.
Answer: The sample space has 6 outcomes: AH, AT, BH, BT, CH, CT.
Example 2: Count outfits
Core
Maria has 3 shirts and 4 pairs of pants. An outfit is one shirt and one pair of pants. How many outcomes are in the sample space of outfits, and what is the probability that a randomly chosen outfit uses her red shirt?
Each of the 3 shirts goes with each of the 4 pants: 3 × 4 = 12 outfits.
The red shirt appears in 4 of them, one with each pair of pants.
P = 412 = 13.
Answer: There are 12 outfits, and the probability of the red shirt is 1/3.
Example 3: Three children
Test-style
A family has 3 children. Assume each child is equally likely to be a boy (B) or a girl (G). What is the probability that exactly 2 of the children are girls?
Sample space: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG, which is 23 = 8 outcomes.
Exactly 2 girls: BGG, GBG, GGB, which is 3 outcomes.
P = 38.
Answer: The probability is 3/8.
Common mistakes
✗ MistakeTreating HT and TH as the same outcome, so two coin flips have only 3 outcomes: 2 heads, 1 head, 0 heads.
✓ InsteadList ordered results: HH, HT, TH, TT. One head happens 2 ways, so P(exactly one head) = 24.
“2 heads, 1 head, 0 heads” are not equally likely; the ordered list is.
✗ MistakeAdding instead of multiplying: 3 shirts and 4 pants make 3 + 4 = 7 outfits.
✓ Instead3 × 4 = 12 outfits.
Every shirt pairs with every pair of pants, which is multiplication.
✗ MistakeUsing the sums 2 to 12 as equally likely outcomes of two dice: P(sum 7) = 111.
✓ InsteadUse the 36 equally likely pairs: 6 of them sum to 7, so P = 636 = 16.
Sums can happen in different numbers of ways, so they are not equally likely.
An event is some of the outcomes, like {2, 4, 6}. The sample space is all of them, {1, 2, 3, 4, 5, 6}.
Quick check: 3 questions
Question 1 of 3
What is the sample space for flipping a coin and rolling a number cube?
Show answer and explanation
Answer: C.2 × 6 = 12 outcomes, from H1 to T6. 6 counts only the cube, 8 adds 2 + 6, and 36 is two number cubes.
Question 2 of 3
Which list is the sample space for flipping a coin twice?
Show answer and explanation
Answer: C. Two flips give four ordered outcomes. {H, T} is one flip, and the other two lists merge HT and TH, which are different outcomes.
Question 3 of 3
A lock code is 3 digits, each 0 to 9. How many outcomes are in the sample space?
Show answer and explanation
Answer: B. Each digit has 10 choices: 10 × 10 × 10 = 1,000. 30 adds, 27 is 33, and 720 forbids repeated digits, which the problem allows.
Sample Space: frequently asked questions
How do you find the sample space?
List every possible outcome in an organized way. For one step, just list the results. For two steps, make a table with one step across the top and the other down the side, or draw a tree diagram. Check your count with multiplication: a coin and a number cube give 2 × 6 = 12 outcomes.
What is the sample space of rolling two dice?
If you record both faces, it is the 36 ordered pairs from (1, 1) to (6, 6), since 6 × 6 = 36, and all are equally likely. If you record only the sum, the sample space is 2 to 12, but those 11 sums are not equally likely; 7 happens 6 ways, 2 only 1 way.
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