Probabilities of Equally Likely Outcomes

The Idea of Equally Likely Outcomes

When you toss a fair coin, you know that heads and tails are equally likely. But what does "equally likely" mean in a precise, mathematical sense? It means that every simple event in the sample space has the same probability of occurring. This is the simplest and most common situation in probability problems, and it leads directly to a clean formula for the probability of any event.

Suppose an experiment has a sample space $S$ containing $n$ distinct outcomes:

If all these outcomes are equally likely, then each simple event $\{\omega_i\}$ must have the same probability. Let that common probability be $p$, where $0 \leq p \leq 1$. So we have:

Now, the sum of the probabilities of all simple events in the sample space must equal 1. This gives:

Since each term is $p$, this sum is $p$ added to itself $n$ times, which is $np$. Therefore:

And so:

This is the fundamental result: when outcomes are equally likely, each individual outcome has probability $1/n$.

From Simple Events to Any Event

Now let $E$ be any event (a subset of the sample space $S$). Suppose the total number of outcomes in $S$ is $n(S) = n$, and the number of outcomes favourable to $E$ is $n(E) = m$. Since each outcome is equally likely, the probability of $E$ is simply the sum of the probabilities of the outcomes that make up $E$.

Each favourable outcome has probability $1/n$, and there are $m$ of them. So:

This gives the classic formula:

This formula only works when all outcomes are equally likely. If they are not, you cannot use it — you would need the more general approach of assigning probabilities individually.

A Quick Example to Fix the Idea …