Probability: How Likely Is It?

The scale from 0 to 1

A probability is a number saying how likely something is.

Every probability lands somewhere on that line:

Probability	Meaning
$0$	cannot happen
$0.25$	unlikely
$0.5$	as likely as not
$0.75$	likely
$1$	certain

Nothing sits outside $0$ and $1$. An answer of $1.4$ or $-0.2$ is not an unlikely event; it is arithmetic that has gone wrong.

Counting equally likely outcomes

When every outcome has the same chance:

$$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$$

A standard die has six faces, all equally likely.

$$P(\text{rolling a } 3) = \frac{1}{6}$$ $$P(\text{rolling an even number}) = \frac{3}{6} = \frac{1}{2}$$ $$P(\text{rolling less than } 5) = \frac{4}{6} = \frac{2}{3}$$

The words “equally likely” are doing real work. This formula applies to a fair die and a fair coin. It does not apply to tomorrow’s weather, where the possible outcomes are not equally likely at all.

The complement: when an event does not happen

Something has to happen, so all the probabilities together add to $1$:

$$P(\text{not } A) = 1 - P(A)$$

If rain has probability $0.3$, then no rain has probability $0.7$.

This is often the quicker route. Finding the probability of rolling at least one six in several rolls is hard head-on and short work through its complement, which is rolling no sixes at all.

Theoretical and experimental probability

Theoretical probability comes from reasoning about the outcomes. A fair coin gives $P(\text{heads}) = \tfrac{1}{2}$, and no coin needs to be flipped to know it.

Experimental probability comes from doing it:

$$P(\text{experimental}) = \frac{\text{times it happened}}{\text{times you tried}}$$

Flip a coin $10$ times and get $7$ heads, and the experimental probability is $\tfrac{7}{10}$.

That is not evidence the coin is unfair. Small numbers of trials vary a lot, and $7$ heads in $10$ is an ordinary result for a perfectly fair coin.

Flips	Heads	Experimental $P$
$10$	$7$	$0.70$
$100$	$56$	$0.56$
$1000$	$508$	$0.508$
$10{,}000$	$4{,}991$	$0.4991$

The experimental value settles towards $0.5$ as trials increase. That tendency has a name — the law of large numbers — and it is why a casino can be confident of its takings over a year while having no idea how one evening will go.

Why the two kinds are both needed

Theoretical probability requires knowing the outcomes and that they are equally likely. For a die or a coin, that is given. For most real questions it is not.

Nobody can reason out the probability that a particular drawing pin lands point up. It depends on its shape and its weight, and there is no symmetry argument to appeal to. The only way to find out is to drop it a thousand times and count — experimental probability is the only method available.

The same is true of how often a bus is late, or how often a machine part fails. Experimental probability turns observed frequencies into a model of a process nobody could reason about from first principles, and that is most of what applied probability does.

Worked examples

Common mistakes

Practice problems