Probability measures the likelihood that an event will occur, expressed as a number between 0 (impossible) and 1 (certain). This guide covers the fundamental formulas and common applications.
Basic Probability Formula
P(Event) = Number of favourable outcomes / Total possible outcomes
Example: Rolling a 4 on a standard die:
P(4) = 1/6 ≈ 0.167 = 16.7%
Example: Drawing a red card from a deck:
P(red) = 26/52 = 1/2 = 50%
Probability as Percentage
Multiply by 100 to convert to percentage:
- P = 0.25 → 25%
- P = 1/3 → 33.3%
- P = 0.05 → 5%
Complementary Probability
The probability that an event does NOT occur:
P(not A) = 1 - P(A)
Example: Probability of NOT rolling a 6:
P(not 6) = 1 - 1/6 = 5/6 ≈ 83.3%
AND Probability (Both Events)
For independent events (one doesn't affect the other):
P(A and B) = P(A) × P(B)
Example: Flipping heads twice in a row:
P(H and H) = 0.5 × 0.5 = 0.25 = 25%
OR Probability (Either Event)
P(A or B) = P(A) + P(B) - P(A and B)
Example: Drawing a King OR a Heart from a deck:
P(King) = 4/52
P(Heart) = 13/52
P(King AND Heart) = 1/52 (King of Hearts)
P(King OR Heart) = 4/52 + 13/52 - 1/52 = 16/52 = 30.8%
Conditional Probability
The probability of A given that B has already occurred:
P(A|B) = P(A and B) / P(B)
Example: A bag has 3 red and 2 blue balls. You draw a red one (not replaced). What's the probability the next is also red?
P(2nd red | 1st red) = 2/4 = 50%
(Only 4 balls remain, 2 are red)
Common Probability Applications
- Weather: 70% chance of rain = 0.7 probability
- Medicine: 95% confidence interval = P(result within range) = 0.95
- Quality control: Defect rate of 0.1% = P(defect) = 0.001
- Gambling: House edge = P(casino wins) slightly above 0.5
Use our Combinations & Permutations Calculator to calculate probabilities for complex counting problems.