Probability Rules: AND, OR, Mutually Exclusive & Independent
Probability Rules
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain). It can also be written as a fraction, decimal or percentage.
Basic Probability
P(event) = number of favourable outcomes ÷ total number of possible outcomes
Worked Example: A bag contains 3 red, 5 blue and 2 green balls. Find P(blue).
- P(blue) = 5/10 = 1/2
Complementary Events
The probability of an event NOT happening:
P(not A) = 1 − P(A)
If P(rain) = 0.3, then P(no rain) = 1 − 0.3 = 0.7
The OR Rule (Addition Rule)
For mutually exclusive events (they cannot happen at the same time):
P(A or B) = P(A) + P(B)
Worked Example: A fair dice is rolled. Find P(2 or 5).
- P(2) = 1/6, P(5) = 1/6
- P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3
For non-mutually exclusive events (Higher):
P(A or B) = P(A) + P(B) − P(A and B)
You subtract P(A and B) because it gets counted twice.
Worked Example: In a class of 30, 18 study French, 12 study Spanish, and 5 study both. Find the probability a randomly chosen student studies French or Spanish.
- P(F or S) = 18/30 + 12/30 − 5/30 = 25/30 = 5/6
The AND Rule (Multiplication Rule)
For independent events (one does not affect the other):
P(A and B) = P(A) × P(B)
Worked Example: A coin is flipped and a dice is rolled. Find P(heads and 6).
- P(heads) = 1/2, P(6) = 1/6
- P(heads and 6) = 1/2 × 1/6 = 1/12
For dependent events (without replacement), the probability of the second event changes:
Worked Example: A bag has 4 red and 6 blue balls. Two are drawn without replacement. Find P(both red).
- P(1st red) = 4/10
- P(2nd red | 1st was red) = 3/9
- P(both red) = 4/10 × 3/9 = 12/90 = 2/15
Mutually Exclusive vs Independent
| Mutually Exclusive | Independent | |
|---|---|---|
| Can happen together? | No | Yes |
| Rule | P(A or B) = P(A) + P(B) | P(A and B) = P(A) × P(B) |
| Example | Rolling a 3 and a 5 on one dice | Coin flip and dice roll |
Common misconception: Mutually exclusive and independent are NOT the same thing. In fact, if two events with non-zero probabilities are mutually exclusive, they CANNOT be independent (knowing one happened means the other definitely did not).
Expected Frequency
Expected frequency = probability × number of trials
Worked Example: A dice is rolled 300 times. How many sixes are expected?
- Expected sixes = 1/6 × 300 = 50
This is a theoretical expectation — actual results may differ due to randomness.
Exam Tips
- "And" usually means multiply; "or" usually means add
- Check whether the question says "with replacement" or "without replacement" — this determines whether events are independent or dependent
- Expected frequency does not have to be a whole number (e.g. 16.7 is acceptable)
- The probabilities of all possible outcomes must sum to 1 — use this to find a missing probability