Probability rules

Why this matters

Probability is the language statistics uses to talk about uncertainty. A quality engineer asks how likely a batch is to contain a defect, an actuary prices the chance of a claim, a data scientist estimates how often a model is wrong. You do not need advanced math for most of it. Four rules, applied carefully, cover a surprising amount of real work.

The basics

An experiment is anything with an uncertain result: rolling a die, testing a part, a customer visiting a page. The sample spaceThe set of all possible outcomes of an experiment. is the set of all possible outcomes. An event is a set of outcomes you care about.

When all outcomes are equally likely:

P(event) = (number of outcomes in the event) / (total number of outcomes)

Rolling a fair six-sided die, P(even) = 3/6 = 0.5, because {2, 4, 6} are three of six outcomes.

Probabilities are always between 0 (impossible) and 1 (certain). You can write them as fractions, decimals or percents: 1/4 = 0.25 = 25%.

When outcomes are not equally likely, use relative frequency: the share of times the event happened in a long record. If 37 of the last 1,000 shipped units were returned, P(return) is about 0.037.

Rule 1: the complement

The probability that an event does not happen is one minus the probability that it does.

P(not A) = 1 - P(A)

This is often the easy way in. "What is the chance of at least one defect in 3 parts?" is hard directly but easy as 1 − P(no defects).

Rule 2: addition (A or B)

P(A or B) = P(A) + P(B) - P(A and B)

You subtract P(A and B) because outcomes in both events were counted twice.

Worked example: draw one card from a standard 52-card deck. What is P(heart or king)?

P(heart) = 13/52
P(king)  = 4/52
P(heart and king) = 1/52     (the king of hearts)
P(heart or king) = 13/52 + 4/52 - 1/52 = 16/52 ≈ 0.308

If two events cannot both happen (they are mutually exclusiveDescribes events that cannot both happen, so P(A and B) = 0.), P(A and B) = 0 and the rule becomes plain addition. Rolling a 1 or a 6 on one die: 1/6 + 1/6 = 1/3.

Rule 3: multiplication for independent events (A and B)

Two events are independentDescribes events where one happening does not change the probability of the other. when one happening does not change the probability of the other. Separate coin flips, separate dice, parts from a stable process.

P(A and B) = P(A) * P(B)        (independent events only)

Worked example: a process makes 2% defective parts, independently. Pick 3 parts.

P(no defect in one part) = 0.98
P(no defects in 3 parts) = 0.98 * 0.98 * 0.98 ≈ 0.941
P(at least one defect)   = 1 - 0.941 = 0.059

That combines the multiplication rule with the complementThe event that A does not happen; its probability is 1 - P(A). rule, the most common pattern in practice.

Rule 4: dependent events

If events are not independent, multiply by the probability of the second event given the first:

P(A and B) = P(A) * P(B given A)

Draw two cards without putting the first back. P(both aces) = 4/52 × 3/51 ≈ 0.0045, because after one ace is gone only 3 aces remain among 51 cards. The next lesson is all about this "given" idea.

Checking with simulation

When a calculation feels slippery, simulate it. Python's random module with a fixed seedA starting value for a random number generator that makes the same "random" results repeat every run. gives the same "random" results every run, which makes results reproducible:

import random

random.seed(42)
trials = 100_000
hits = 0
for _ in range(trials):
    a, b = random.randint(1, 6), random.randint(1, 6)
    if a == 6 or b == 6:
        hits += 1
print(hits / trials)   # close to 11/36 ≈ 0.306

The exact answer: P(at least one six in two rolls) = 1 − (5/6)² = 11/36 ≈ 0.306. The simulation agrees to two or three decimal places, and more trials bring it closer.

Common mistakes

• Adding probabilities of events that overlap without subtracting the overlap.
• Multiplying probabilities of events that are not independent.
• Treating "three heads in a row" as making tails "due". Independent flips have no memory: this is the gambler's fallacyThe mistaken belief that past independent results, such as three heads, make another result "due"..
• Getting a probability above 1 or below 0. That is always a sign of an error.

Quick recap

• Equally likely outcomes: favourable / total.
• Complement: P(not A) = 1 − P(A).
• Addition: P(A or B) = P(A) + P(B) − P(A and B).
• Multiplication: P(A and B) = P(A) × P(B) when independent; otherwise use P(B given A).
• Simulate with a fixed seed to check your reasoning.

Try it

In the Do tab, write the addition rule and compare it with a simulation.