Probability Basics: A Practical Introduction

Outcomes, events and the probability scale

Probability starts with an experiment, such as rolling a die, and its sample space, the set of all possible outcomes. For one roll of a standard die the sample space is $\{1, 2, 3, 4, 5, 6\}$. An event is any collection of outcomes, such as rolling an even number, which is the set $\{2, 4, 6\}$.

Every probability lies between 0 and 1. A probability of 0 means the event cannot happen, 1 means it is certain, and 0.5 means it is as likely to happen as not. Probabilities can be written as fractions, decimals or percentages, so $\frac{1}{4}$, 0.25 and 25 percent all mean the same thing. When all outcomes are equally likely, the probability of an event is a simple ratio:

So the probability of rolling an even number is $\frac{3}{6} = \frac{1}{2}$. With two dice there are $6 \times 6 = 36$ equally likely ordered outcomes, and six of them give a sum of 7, namely (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1), so $P(\text{sum} = 7) = \frac{6}{36} = \frac{1}{6}$. Counting outcomes in bigger problems uses the techniques in permutations and combinations.

The complement rule

The complement of an event $A$ is everything that is not in $A$. Since one or the other must happen, their probabilities add to 1:

Whenever a question says at least one, try the complement first. It almost always turns a long calculation into a short one.

The addition rule: A or B

The probability that $A$ or $B$ happens, or both, is found by adding their probabilities and subtracting the overlap, which would otherwise be counted twice:

Drawing one card from a standard 52-card deck, the probability of a heart or a king is $\frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$, because the king of hearts belongs to both groups. Events that cannot happen together, such as rolling a 2 and rolling a 5 on a single roll, are mutually exclusive. Their overlap is zero, so their probabilities simply add.

The multiplication rule: A and B

Two events are independent when the outcome of one does not change the probability of the other, as with separate coin flips. For independent events, multiply:

Two heads in two flips therefore has probability $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$. When events are dependent, the second probability must reflect what has already happened. Drawing two cards without replacement, the probability that both are aces is $\frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$, because after one ace is removed only 3 aces remain among 51 cards.

Conditional probability and Bayes' theorem

Conditional probability is the probability of $A$ given that $B$ has happened, written $P(A \mid B)$. Knowing $B$ shrinks the sample space to $B$ alone:

Bayes' theorem reverses a conditional probability, turning $P(B \mid A)$ into $P(A \mid B)$. Its most famous application shows why a positive medical test can be less alarming than it sounds.

Expected value

The expected value of a game or random quantity is the long-run average result, found by multiplying each outcome by its probability and adding:

Suppose a game pays 10 dollars if a die shows six and costs you 2 dollars otherwise. The expected value is $10 \cdot \frac{1}{6} - 2 \cdot \frac{5}{6} = \frac{10 - 10}{6} = 0$, so the game is fair: over many plays you would expect to break even. Casino games are designed with a slightly negative expected value for the player, which is how casinos profit over millions of bets even though individual players sometimes win. Insurance works on the same principle from the other side: the insurer collects slightly more in premiums than it expects to pay out in claims.

Common traps

• The gambler's fallacy: after five reds on a fair roulette wheel, black is not due. Each spin is independent, so the probability is unchanged.
• Adding probabilities of events that overlap without subtracting the overlap, which can produce probabilities greater than 1.
• Multiplying probabilities of events that are not independent.
• Confusing $P(A \mid B)$ with $P(B \mid A)$, as in the medical test example.
• Underestimating coincidences: in a group of only 23 people, the chance that two share a birthday is about 50.7 percent.

The law of large numbers explains what probability does promise: as an experiment is repeated many times, the proportion of times an event occurs settles close to its probability. It makes no promise about any short run of results. The probability calculator handles single, combined and conditional events with steps, and the permutation and combination calculator counts the outcomes that many probability problems need.

Tree diagrams for multi-step problems

When an experiment happens in stages, a tree diagram keeps track of every path. Each branch is labelled with its probability, you multiply along a path to get the probability of that whole sequence, and you add the paths that make up the event you want.

Tree diagrams make the multiplication rule visual and show clearly where probabilities change between stages. They are also the most reliable way to set up a Bayes' theorem problem, because each branch of the second stage is a conditional probability.

Frequently asked questions