Naive Probability¶

Definition¶

Let $S$ be a finite sample space whose outcomes are equally likely. For an event $A\subseteq S$ $S$ , the naive probability of $A$ is

\mathbb{P}(A)=\frac{|A|}{|S|}.

Interpretation¶

When a finite sample space has equally likely outcomes, the probability of an event is the share of outcomes that belong to it. Probability then becomes counting: count the favourable outcomes, count all outcomes, and divide.

A 6 by 6 grid of all pairs of two dice with the six pairs that sum to 7 shaded, giving probability 6/36 = 1/6.

The grid lists all $36$ ordered pairs $(i,j)$ , with $i$ the first die and $j$ the second; for fair dice each cell is equally likely. The six shaded cells $(1,6),(2,5),\ldots,(6,1)$ have sum $7$ , so $\mathbb{P}(\text{sum}=7)=\tfrac{6}{36}=\tfrac{1}{6}$ $\frac{1}{6}$ . The pairs $(1,6)$ and $(6,1)$ are different cells: recording which die shows which number is what makes all outcomes equally likely.

The equal-likelihood assumption belongs to the model, not to the question, and the choice of outcomes decides whether it holds. Taking the eleven possible sums $2,\ldots,12$ as outcomes and dividing by $11$ gives $\mathbb{P}(\text{sum}=7)=\tfrac{1}{11}$ $\frac{1}{11}$ , which is wrong, because a sum of $7$ arises in six ways and a sum of $2$ in only one. The sample space must also be finite: infinitely many outcomes cannot all have the same positive probability, since their probabilities would add up to more than $1$ .

The formula satisfies the axioms of probability, because the number of elements (cardinality) of a union of disjoint sets is the sum of their numbers of elements. The multiplication rule, permutations and combinations are the tools for computing $|A|$ and $|S|$ .

Key Results¶

Naive probability applies only when all outcomes in the sample space are equally likely.
It is computed as favourable outcomes divided by total outcomes.
Counting errors in either numerator or denominator lead directly to probability errors.

Examples¶

The probability of rolling an even number on a fair die is three favourable faces out of six.
The probability of drawing an ace from a well-shuffled deck is four favourable cards out of fifty-two.
If a random PIN is chosen uniformly from all four-digit PINs, naive probability can compute the chance of a repeated digit pattern.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 1

Aliases: classical probability, Laplace probability, naive definition of probability, classical definition of probability Keywords: equally likely outcomes Symbols: subset

Last updated: 20 October 2026 03:02