Probability Basics
How to calculate simple probabilities, including combined events, with worked examples.
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, decimal or percentage.
The basic formula
When all outcomes are equally likely:
P(event) = number of favourable outcomes ÷ total number of outcomes.
Example. A fair six-sided die is rolled. The probability of rolling an even number is 3/6 = 1/2, because 2, 4 and 6 are favourable out of six outcomes.
Complementary events
P(not A) = 1 − P(A).
If the probability of rain is 0.3, the probability of no rain is 0.7.
Independent events
Two events are independent if one does not affect the other. For independent events:
P(A and B) = P(A) × P(B).
Example. Toss a fair coin and roll a fair die. P(heads and a 6) = 1/2 × 1/6 = 1/12.
Mutually exclusive events
Events that cannot happen together are mutually exclusive. Then:
P(A or B) = P(A) + P(B).
Example. Roll a die. P(1 or 2) = 1/6 + 1/6 = 1/3.
Without replacement
A bag has 3 red and 2 blue counters. Two are drawn without putting the first back. P(both red):
- First red: 3/5.
- Second red: 2/4 = 1/2.
- Both: 3/5 × 1/2 = 3/10.
Common mistakes
- Adding probabilities for events that are not mutually exclusive.
- Forgetting that the total changes when items are not replaced.
- Giving a probability greater than 1 or less than 0.
- Thinking that after several heads, tails is "due". For a fair coin, each toss is independent.
Quick recap
Count favourable over total, use 1 − P(A) for "not", multiply for independent "and", add for mutually exclusive "or".