Probability is the branch of mathematics that gives a precise way to talk about how likely events are, from flipping a coin to forecasting rain. Understanding the basics helps you make better decisions and see through misleading statistics.
The Basic Formula
For equally likely outcomes, probability is the number of favourable outcomes divided by the total number of possible outcomes, expressed as a number between 0 (impossible) and 1 (certain).
Probability = Favourable outcomes ÷ Total possible outcomes
Complementary probability = 1 − Probability of the event
Common Probability Examples
| Scenario | Calculation | Probability |
|---|---|---|
| Rolling a specific number on a die | 1 ÷ 6 | 16.7% |
| Flipping heads | 1 ÷ 2 | 50% |
| Two heads in a row (independent) | 0.5 × 0.5 | 25% |
| Drawing an ace from a deck | 4 ÷ 52 | 7.7% |
| At least one 6 in three rolls | 1 − (5/6)³ | 42.1% |
Independent vs Dependent Events
Two events are independent if the outcome of one does not affect the other — flipping a coin twice produces independent events. To find the probability of two independent events both occurring, multiply their individual probabilities.
Dependent events are those where one outcome affects the other. Drawing two cards without replacement is dependent — after the first card, the deck has 51 cards remaining, changing the probability of the second draw.
Probability in Everyday Life
- Weather forecasts: A 70% chance of rain means it rained roughly 7 times out of 10 under similar past conditions
- Medical tests: A 95% sensitive test still produces false negatives in 5% of people who have the condition
- Insurance pricing: Premiums are based on the probability of a claim, calculated from millions of policyholders
Common Probability Mistakes
The gambler's fallacy is the false belief that past random events affect future ones — if a coin lands heads ten times in a row, the probability of heads on the eleventh flip is still exactly 50%, because the coin has no memory of previous flips. This fallacy drives a remarkable amount of poor decision-making in gambling, investing, and everyday life, where people assume a "streak" makes a reversal more likely.
Ignoring base rates is another common error, and arguably more consequential. If a rare disease affects 1% of the population and a test for it is 99% accurate, a positive result does not mean you almost certainly have the disease. The maths of conditional probability show that, because the disease is rare to begin with, a meaningful proportion of all positive results will be false positives — even with a highly accurate test. This is why doctors interpret single test results in the context of base rates rather than treating accuracy figures in isolation.
The gambler's fallacy is the false belief that past random events affect future ones — if a coin lands heads ten times in a row, the probability of heads on the eleventh flip is still exactly 50%. Ignoring base rates is another error: if a rare disease affects 1% of people and a test is 99% accurate, a positive result still does not mean you almost certainly have the disease, because the maths of conditional probability produces many false positives when the underlying condition is rare.
Conditional Probability
Conditional probability describes the likelihood of an event occurring given that another event has already happened, and it's where a great deal of everyday probabilistic reasoning goes wrong. The probability of rain given that the sky is currently overcast is different from the unconditional probability of rain on any random day, because the overcast sky provides genuine information that updates the estimate. Formally, conditional probability is written as P(A|B), read as "the probability of A given B."
This concept underlies Bayes' theorem, a formula used extensively in medical diagnosis, spam filtering, and forensic science to update probability estimates as new evidence arrives. A practical example: if 1% of people have a certain condition, and a test correctly identifies 95% of true cases while also incorrectly flagging 5% of healthy people, the conditional probability that someone who tests positive actually has the condition is far lower than 95% — because the much larger pool of healthy people generates many false positives even at a 5% error rate. Working through the actual numbers consistently surprises people who haven't encountered this reasoning before, which is exactly why understanding conditional probability is so valuable for interpreting real-world test results correctly.