Probability & Counting · Grades 8–11
Probability Basics: One Event, “And” and “Or”
One event
If every outcome is equally likely, P = favourable ÷ total. Thirteen hearts in 52 cards gives 1/4 = 25%.
Drawing a heart
- Decimal: 0.25
- Odds against: 39 to 13
Step-by-step working 3 steps
- 1Favourable ÷ total
P = 13 / 52 = 1/4
Assumes every outcome is equally likely.
- 2As decimal and percent
0.25 = 25%
- 3Probability of NOT happening
1 − 0.25 = 0.75
Both events (independent)
Multiply the probabilities. Two heads in a row: 25%.
Two heads in a row
- Decimal: 0.25
Step-by-step working 2 steps
- 1Read the two probabilities
P(A) = 0.5, P(B) = 0.5
These formulas assume A and B are independent: one does not change the chance of the other.
- 2Multiply
P(A and B) = P(A) × P(B) = 0.5 × 0.5 = 0.25
Either event
Add the probabilities and subtract the overlap. For P(A) = 50% and P(B) = 1/6: 58.3333%.
A or B
- Decimal: 0.583333
Step-by-step working 2 steps
- 1Read the two probabilities
P(A) = 0.5, P(B) = 0.166667
These formulas assume A and B are independent: one does not change the chance of the other.
- 2Add, then remove the overlap
P(A or B) = P(A) + P(B) − P(A and B) = 0.5 + 0.166667 − 0.083333 = 0.583333
Without subtracting the overlap, outcomes where both happen would be counted twice.
Check independence first
Common mistakes
Multiplying dependent events
The product rule needs independence.
Forgetting the overlap
For “or”, subtract P(A and B).
Assuming equal likelihood
favourable ÷ total only works when outcomes are equally likely.
Frequently asked questions
Can a probability be above 1?
No. Probabilities run from 0 (impossible) to 1 (certain).
