No gaps, no repeats

The idea

The sample space S must include every possible outcome, list no outcome more than once, and be detailed enough for the question; its number of elements is the sample size n(S).

Build a careful list

Two coins are tossed together. Is HT the same outcome as TH? And how much detail should a list have if we only care about rain?

How many outcomes are there when two coins are tossed?

• 3
• 4
• 2

Complete, no repeats

Tap every outcome that belongs in the sample space, then Check. The list must include every outcome, list none twice and be detailed enough.

Rules for the list

The sample space S must include every possible outcome, list no outcome more than once, and be detailed enough for the question. The number of elements is the sample size n(S).

Two coins: S = {HH, HT, TH, TT} and n(S) = 4. If we only ask "rain or not", S = {Rain, No Rain} and n(S) = 2. If we want the amount of rain, we need a longer list.

Notes

The sample space S has every possible outcome, none twice, with enough detail for the question. Its number of elements is the sample size n(S).

Check yourself

Two coins are tossed together. What is the sample size n(S)?

Answer: 4

S = {HH, HT, TH, TT}, so n(S) = 4. HT and TH are different outcomes.

A team plays a match. The sample space is S = {Win, Lose, Draw}. What is n(S)?

Answer: 3

There are three elements, so n(S) = 3.

Which list is not a good sample space for rolling a die?

{Rain, No Rain} is enough when we only ask: "Will it rain?" What if we also want to know the amount of rain?

• {1, 2, 3, 4, 5, 6}. This is the correct sample space.
• {1, 2, 3, 4, 5} — correct. Right. It leaves out 6, so it does not include every outcome.
• {6, 5, 4, 3, 2, 1}. The order of listing does not matter. It has every outcome once.
• {Rain, No Rain} is still enough. It cannot tell drizzle from heavy rain.
• We should list Rain twice. No outcome should be listed more than once.
• We need a longer list, such as {No Rain, Drizzle, Light Rain, Heavy Rain} — correct. Yes. The sample space must be detailed enough for the question.