Outcomes - The Possible Results of an Experiment

An outcome is one possible result of a single trial of a probability experiment. When you roll a die, each of the numbers 1, 2, 3, 4, 5, and 6 is a separate outcome. Listing every possible outcome clearly and completely is the starting point for every probability calculation.

Systematic counting of outcomes is far older than formal probability theory: the ancient Chinese I Ching, dating back roughly 3,000 years, is built entirely from 64 hexagrams – every possible combination of six stacked lines that can each be one of two types, which is exactly 2 × 2 × 2 × 2 × 2 × 2 = 64 outcomes from the multiplication principle, discovered millennia before anyone wrote the rule down formally. The same multiplication principle governs digital security today: a 4-digit PIN has 10 × 10 × 10 × 10 = 10,000 possible outcomes, which is precisely why banks combine a PIN with a physical card – 10,000 combinations alone would be far too easy to guess by trial and error.

Elementary Outcomes

An elementary outcome (also called a simple outcome or sample point) is an individual result that cannot be broken down any further. For example, rolling a 3 on a die is a single elementary outcome – you cannot split it into smaller results.

Equally Likely Outcomes

Outcomes are equally likely when each one has exactly the same chance of occurring. A fair coin has two equally likely outcomes: Heads and Tails. A fair die has six equally likely outcomes: 1, 2, 3, 4, 5, 6. When outcomes are equally likely, theoretical probability is straightforward to calculate.

Listing Outcomes Systematically

For experiments with two or more steps, it helps to list outcomes in a systematic way to make sure none are missed. Common methods include:

Ordered lists
Two-way tables (grids)
Tree diagrams (covered in a later topic)

Worked Examples

List all outcomes when a fair coin is tossed twice.

First toss	Second toss	Outcome
H	H	HH
H	T	HT
T	H	TH
T	T	TT

Total outcomes = 4.

List all outcomes when a die is rolled and a coin is tossed at the same time.

Die outcomes: 1, 2, 3, 4, 5, 6. Coin outcomes: H, T. Combine each:

(1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H), (4,T), (5,H), (5,T), (6,H), (6,T)

Total outcomes = 6 × 2 = 12.

A bag contains one Red, one Blue, and one Green ball. One ball is drawn and not replaced; then a second is drawn. List all outcomes.

(R,B), (R,G), (B,R), (B,G), (G,R), (G,B) – Total = 6 outcomes.

Note: (R,R) is impossible because the ball is not replaced.

Counting Outcomes: The Multiplication Principle

If experiment A has m outcomes and experiment B has n outcomes, then doing both together has m × n outcomes. This extends to any number of experiments multiplied together.

Key Takeaways

An outcome is one specific result of a probability experiment.
List outcomes systematically – use tables or diagrams to avoid missing any.
Equally likely outcomes simplify probability calculation.
For multi-step experiments, total outcomes = product of individual outcome counts.

Practice: Counting Outcomes

The Multiplication Principle