Probability — Summary

1 Basic Concepts

Term	Meaning	Example (rolling a die)
Experiment	A process that produces random results	Rolling a 6-sided die
Outcome	A single possible result	Rolling a 4
Sample space (S)	The set of all possible outcomes	S = {1, 2, 3, 4, 5, 6}
Event (A)	A set of one or more outcomes	A = {rolling an even} = {2, 4, 6}
Favourable outcomes	Outcomes that belong to the event	3 outcomes: 2, 4, 6

Probability Scale

P(A) = 0 → impossible event
P(A) = 1 → certain event
0 < P(A) < 1 → possible but not certain
All probabilities are between 0 and 1 (or 0% and 100%)

2 Theoretical Probability

Based on reasoning and counting outcomes — assumes all outcomes are equally likely.

Theoretical P(A)

P(A) = number of favourable outcomes / total outcomes in S

✏️

Rolling a fair die — P(rolling a number greater than 4)?
Favourable outcomes: {5, 6} → 2 outcomes
Total outcomes: {1,2,3,4,5,6} → 6
P(A) = 2/6 = 1/3 ≈ 33.3%

✏️

A bag has 3 red, 5 blue, 2 green marbles. P(blue)?
Total = 3 + 5 + 2 = 10 marbles
P(blue) = 5/10 = 1/2 = 50%

3 Experimental Probability

Based on actual results from conducting an experiment (trials).

Experimental P(A)

P(A) = number of times A occurred / total number of trials

✏️

A coin is flipped 50 times. Heads appears 28 times. Experimental P(heads)?
P(heads) = 28/50 = 0.56 = 56%
(Theoretical probability = 0.5 = 50%)

Theoretical vs Experimental

	Theoretical	Experimental
Based on	Logic and equally-likely outcomes	Actual data from trials
Formula	favourable / total	successes / trials
Exact?	Yes (for fair, simple experiments)	Varies — depends on how many trials
Improves with	Better counting	More trials (Law of Large Numbers)

🔑Law of Large Numbers: As the number of trials increases, the experimental probability gets closer to the theoretical probability.

4 Complementary Events

The complement of event A (written A' or Ā) is the event that A does not happen.

Complement rule

P(A') = 1 − P(A)

Together

P(A) + P(A') = 1 (always)

✏️

P(rain tomorrow) = 0.35. P(no rain)?
P(no rain) = 1 − 0.35 = 0.65

💡Use the complement when it's easier to count what you don't want, then subtract from 1.

5 Tree Diagrams & Tables

Tree Diagrams

Used to list all outcomes of multi-step experiments. Each branch represents one outcome. Multiply along branches, add between branches.

✏️

Flip a coin, then roll a die. P(heads and 3)?
Branch 1: Heads (prob = 1/2) → Branch 2: roll 3 (prob = 1/6)
P(heads AND 3) = 1/2 × 1/6 = 1/12
Total outcomes in tree = 2 × 6 = 12

Tables (Two-Way / Outcome Tables)

Useful for two simultaneous events. List outcomes of event 1 in rows, event 2 in columns.

✏️

Rolling two dice — P(sum = 7)?
Make a 6 × 6 table. Pairs that give sum 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6 pairs
Total outcomes = 36
P(sum = 7) = 6/36 = 1/6

6 Common Mistakes to Avoid

Mistake	What to do instead
Probability greater than 1	Probability is always between 0 and 1. If you get > 1, you made an error.
Not listing all outcomes	Be systematic — use a tree diagram or table to make sure you've captured all outcomes.
Confusing theoretical and experimental	Theoretical: equally-likely outcomes formula. Experimental: count actual results from trials.
Forgetting the complement	P(at least one) is often easier as 1 − P(none).
Adding when should multiply	For independent events happening together (AND), multiply probabilities. For either/or (OR), add (if mutually exclusive).
Missing sample space items	Count carefully — for two dice, total outcomes = 36, not 12.