Sample Space and the Counting Principle

Sample Space and the Counting Principle — Moosa Academy

Before finding a probability, you need to know how many outcomes are possible. The sample space lists them, and the fundamental counting principle counts them quickly by multiplying the choices at each step.

Sample space All the possible outcomes

The sample space is the set of all possible outcomes of an experiment. The probabilities of all its outcomes add to 1, and nothing outside it can happen.

Experiment	Sample space	Count
One coin	{H, T}	2
Two coins	{HH, HT, TH, TT}	4
One die	{1, 2, 3, 4, 5, 6}	6
Two dice	all ordered pairs	36

Counting The fundamental counting principle

If an experiment happens in independent steps, the total number of outcomes is the product of the choices at each step.

N = n_1 \times n_2 \times n_3 \times \cdots

Each $n_i$ is the number of outcomes at step $i$ . We multiply because every outcome of one step pairs with every outcome of the next.

A coin then a die: $2 \times 6 = 12$ outcomes — not $2 + 6$ .

Example Counting in action

A die rolled 6 times

Six independent steps, each with 6 outcomes:

6^{6}

⟹ 46,656 outcomes.

A 4-digit PIN (digits 0–9)

Four positions, each with 10 choices:

10^{4}

⟹ 10,000 different PINs.

A menu: 3 appetizers × 5 mains × 4 desserts

3 \times 5 \times 4

⟹ 60 different meals.

A licence plate: 3 letters + 3 digits

Letters have 26 choices each, digits 10 each:

26^{3} \times 10^{3}

⟹ 17,576,000 different plates.

Reference Counting beats listing

Experiment	By counting	Result
Two coins	2 × 2	4
A die twice	6 × 6	36
A die 6 times	6⁶	46,656
A die 10 times	6¹⁰	60,466,176

Summary

The sample space is every possible outcome of an experiment, and its probabilities add to 1.
Counting principle: $N = n_1 \times n_2 \times \cdots$ — multiply the number of outcomes at each step.
We multiply (not add) because each outcome pairs with every outcome of the next step.
It saves enormous effort — millions of outcomes found with a single multiplication.