Probability
Count what could happen, count what you want
Probability is a fraction: favourable outcomes over total possible outcomes. The key is counting carefully.
See It in Action
The Problem
A bag contains 4 red, 3 green, and 5 blue marbles. One is drawn randomly. What is P(not green)?
Common Mistake
P(not green) = 4 + 5 = 9. So the probability is 9.
Probability must be a fraction. The student forgot to write it as a fraction over the total. Also, 9 > 1 which is impossible for a probability.
Correct Approach
Total marbles = 4 + 3 + 5 = 12. Favourable outcomes (not green) = 4 + 5 = 9. P(not green) = 9/12 = 3/4.
Answer: 3/4
P(green) = 3/12 = 1/4. P(not green) = 1 − 1/4 = 3/4. ✓ Both methods agree.
The Core Concept
Probability measures how likely something is to happen. It's always written as a fraction (or decimal or percentage) between 0 (impossible) and 1 (certain). The formula is: P(event) = number of favourable outcomes ÷ total number of equally likely outcomes.
The hardest part is counting correctly. Total outcomes means ALL possible outcomes, including the favourable ones. If a bag has 3 red and 5 blue marbles, P(red) = 3/8 (not 3/5) because total outcomes = 3 + 5 = 8.
For combined events (two coins, two dice), list all possibilities systematically in a table or tree diagram. This prevents double-counting and missing outcomes. Two dice have 6 × 6 = 36 total outcomes, not 12.
Common Mistakes to Avoid
Using only the non-favourable outcomes as the denominator
P(red) = 3/(5+3) doesn't make sense. The denominator is always the TOTAL number of outcomes, which includes the red marbles too: 3/8.
Forgetting that probability must be between 0 and 1
If your answer is greater than 1, something is wrong. Probability can be 0 (impossible) to 1 (certain) and nothing outside that range.
Adding probabilities when they should be multiplied
P(heads on coin 1 AND heads on coin 2) = 1/2 × 1/2 = 1/4. Not 1/2 + 1/2 = 1. "AND" → multiply. "OR" → add (but be careful about overlap).
Not listing all outcomes for combined events
Two spinners with 3 sections each have 3 × 3 = 9 total outcomes, not 6. Always draw a grid or tree diagram for combined events.
Try It Yourself
A spinner has sections numbered 1, 2, 3, 4, 5, 6. What is the probability of spinning a prime number?
Hint: First, which numbers from 1 to 6 are prime? (Remember: 1 is NOT a prime number.)
Key Tips
Always write probability as a fraction: favourable ÷ total. Check it's between 0 and 1.
For "not X": P(not X) = 1 − P(X). Often quicker than counting the non-X outcomes directly.
For combined events, draw a 2D grid to list all outcomes without missing any.
Related Skills
Ready to practise Probability?
Reading is the start. The Maths Gym has exercises designed around this skill — with instant feedback and progress tracking.