Maths⏱ 6 min read

How to Calculate Probability: From Basics to Multiple Events

Probability powers everything from insurance pricing to medical testing to card games. Here's how to calculate it correctly, including the tricky cases where most people go wrong.

Probability is the mathematics of uncertainty. Once you understand the core rules, you can calculate the likelihood of almost anything — and spot when statistics are being used to mislead you.

Basic Probability

Probability = Favourable outcomes ÷ Total possible outcomes P ranges from 0 (impossible) to 1 (certain) To express as percentage: multiply by 100

Example: A standard die has 6 faces. What's the probability of rolling a 4?

P(4) = 1 ÷ 6 = 0.167 = 16.7%

AND Probability (Both Events Occur)

For two independent events (where one doesn't affect the other), the probability of both occurring is the product of their individual probabilities:

P(A AND B) = P(A) × P(B) P(rolling a 4 AND flipping heads) = 1/6 × 1/2 = 1/12 = 8.3%

OR Probability (Either Event Occurs)

For the probability that at least one of two events occurs:

P(A OR B) = P(A) + P(B) − P(A AND B) P(rolling a 4 OR rolling a 6) = 1/6 + 1/6 − 0 = 2/6 = 33.3% (Subtract the AND term to avoid double-counting events where both could happen simultaneously)

Complement Rule

Sometimes it's easier to calculate the probability of something NOT happening, then subtract from 1:

P(event happens) = 1 − P(event doesn't happen) P(rolling at least one 6 in three dice rolls) = 1 − P(no 6 in any roll) = 1 − (5/6 × 5/6 × 5/6) = 1 − 0.579 = 42.1%

Conditional Probability

This is where most intuitions go wrong. Conditional probability asks: given that one event has already happened, what's the probability of another?

P(A given B) = P(A AND B) ÷ P(B)

Classic example (medical testing): A test for a disease is 99% accurate. The disease affects 1% of the population. You test positive. What's the probability you actually have the disease?

Most people say 99%. The actual answer is only about 50%. Here's why:

In 10,000 people: • 100 have the disease: 99 test positive (true positive), 1 missed • 9,900 don't have it: 99 test positive anyway (false positive) Total positive tests: 198 True positives: 99 P(disease | positive test) = 99 ÷ 198 = 50%

This is called the base rate fallacy — ignoring how rare the condition is. It's why screening programmes for rare diseases require very high specificity to be useful, and why a positive test result is always followed by confirmatory testing.

The Gambler's Fallacy

After a coin lands heads 10 times in a row, most people feel tails is "due." It isn't. Each flip is independent — the probability of tails is still exactly 50%.

The confusion arises because the probability of 10 heads in a row before you start flipping is genuinely tiny (1 in 1,024). But once 9 heads have occurred, you're not asking about the whole sequence anymore — you're asking about a single coin flip, which is always 50/50.

Expected Value

Expected value (EV) is the average outcome if you repeated an event many times. It's the foundation of rational decision-making under uncertainty:

EV = Sum of (Outcome × Probability of that outcome) Lottery ticket: costs £2, 1-in-1,000,000 chance of £1,000,000 EV = (1,000,000 × 0.000001) − 2 = £1 − £2 = −1 Negative EV = statistically a losing bet over time

All casino games and most lotteries have negative expected value by design — that's how the house always profits over time. Insurance has negative EV too (you pay more in premiums than you statistically expect to claim), but you buy it for the risk management, not the expected return.

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