Sample spaces

All the possibilities

The sample space is the set of all possible outcomes. Listing it carefully is the first step in finding any probability.

Sample spaces for two events

For a compound event (two or more things at once), use one of these tools.

Organized list

Flip two coins: HH, HT, TH, TT. That’s 4 outcomes. An organized list works through systematically: list all with the first coin H, then all with the first coin T.

Table

Roll two dice. A table (grid) shows all 36 outcomes:

Each cell is one outcome, here showing the sum.

Tree diagram

A tree diagram branches for each stage. Flip a coin, then roll a die:

• H → 1, 2, 3, 4, 5, 6
• T → 1, 2, 3, 4, 5, 6

That’s 2 × 6 = 12 outcomes: H1, H2, … T6.

Counting outcomes quickly

Multiply the number of outcomes at each stage:

• 2 coins: 2 × 2 = 4
• 3 coins: 2 × 2 × 2 = 8
• 2 dice: 6 × 6 = 36
• 3 shirts and 4 pants: 3 × 4 = 12 outfits

Using the sample space for probability

From the two-dice table:

• P(sum = 7) = 6/36 = 1/6 (7 is the most common sum)
• P(sum = 12) = 1/36
• P(doubles) = 6/36 = 1/6

Worked examples

Example 1: List the sample space for flipping a coin and spinning a spinner with A, B, C.

H-A, H-B, H-C, T-A, T-B, T-C → 6 outcomes.

Example 2: How many outcomes for 4 coins?

2⁴ = 16.

Example 3: Two dice: P(sum is 10)?

(4,6), (5,5), (6,4) → 3/36 = 1/12.

Common mistakes

• Treating HT and TH as the same outcome. They’re different.
• Missing outcomes by not listing systematically.
• Adding instead of multiplying stage counts.

Try it yourself

1. How many outcomes for rolling a die and flipping a coin?
2. List the outcomes for two coins.
3. Two dice: P(sum = 2)?
4. Two dice: P(sum is greater than 9)?

Answers: 1) 12 2) HH, HT, TH, TT 3) 1/36 4) 6/36 = 1/6

Quick recap

• The sample space lists every outcome.
• Use an organized list, a table or a tree diagram.
• Multiply stage counts to find the total.