The Mathematics of Maybe: Introduction to Probability · Lesson 5 of 7
Sample Spaces and Events
“List every possible outcome first—probability dislikes missing guests.”
• Construct a complete sample space. • Understand sample size n(S). • Define events as subsets of a sample space. • Distinguish outcomes from events. • Choose the right level of detail for a sample space.
Probability becomes much easier to understand when we first identify all the outcomes that could possibly occur in a situation. Instead of thinking about one result at a time, we begin by listing every possible result in an organised way. This complete collection of possible outcomes is called the sample space.
Once the sample space is known, we can focus on the particular outcome or group of outcomes we are interested in. Such a collection is called an event. In this way, the sample space tells us everything that can happen, while an event tells us what we want to study within those possibilities.
The set of all possible outcomes of a random experiment, usually denoted by S.
The number of elements in the sample space, denoted by n(S).
A sample space must include every possible outcome, and no outcome should be listed more than once.
| Experiment | Sample space | n(S) |
|---|---|---|
| Toss one coin | {H,T} | 2 |
| Roll one die | {1,2,3,4,5,6} | 6 |
| Toss two coins | {HH,HT,TH,TT} | 4 |
| Match result | {Win,Lose,Draw} | 3 |
Why HT and TH Are Different Outcomes
When two coins are tossed, HT means the first coin gives Heads and the second gives Tails. TH means the first gives Tails and the second gives Heads. They are different complete outcomes, so both must appear in the sample space.
The Sample Space Depends on the Question
If our weather question is simply 'Will it rain?', then S={Rain, No Rain} may be enough. If we care about rainfall intensity, a more useful sample space could be {No Rain, Drizzle, Light Rain, Heavy Rain}. The right sample space must match the detail required by the problem.
An event is one outcome or a collection of outcomes from the sample space. In set language, an event is a subset of the sample space.
Problem
Two coins are tossed. Write the event 'at least one Head'.
- 1.Sample space S={HH,HT,TH,TT}.
- 2.The outcomes containing at least one H are HH, HT and TH.
- 3.So E={HH,HT,TH}.
Problem
A standard die is rolled. Write the event 'number greater than 4'.
- 1.S={1,2,3,4,5,6}.
- 2.Only 5 and 6 are greater than 4.
- 3.So E={5,6}.
Problem
A stall offers Samosa, Pakora or Bhaji, and either Chai or Lassi. What are all snack-drink combinations?
- 1.Pair each snack with each drink.
- 2.S={(Samosa,Chai),(Samosa,Lassi),(Pakora,Chai),(Pakora,Lassi),(Bhaji,Chai),(Bhaji,Lassi)}.
- 3.There are 6 possible combinations.
Practice Problems
- Write the sample space when a die is rolled and a coin is tossed together.
- List all integers from −4 to +4 as a sample space and state n(S).
- A box contains green and red balls. One ball is drawn and only its colour is recorded. Write the sample space.
- Two coins are tossed. Write the event 'exactly one Head'.
- A fair die is rolled. Write the event 'an even number'.
- A café offers 2 sandwiches and 3 drinks. List the sample space of all one-sandwich-one-drink combinations.
Key Takeaways
• The sample space contains every possible outcome exactly once. • n(S) is the number of outcomes in the sample space. • An event is a subset of the sample space. • A single outcome and an event are not always the same thing. • The sample space must match the level of detail required by the question.
Next, we use tree diagrams to organise sample spaces for multi-step experiments.