Equally Likely Events and Sample-Point Counting

The outcomes of a random experiment are called equally likely when every individual outcome has exactly the same chance of occurring as every other outcome — no outcome is favoured over any other. This is the assumption behind describing a coin as "fair" or "unbiased," a die as "fair," and a deck of cards as "well-shuffled": each assumes every possible outcome (H or T; 1 through 6; any one of the 52 cards) is exactly as likely as any other.

This assumption is not automatic — it must genuinely hold before the classical probability formula (§5) can be applied by simply counting outcomes. A coin that is known to be biased (say, weighted so it lands Heads more often than Tails) does NOT have equally likely outcomes, and its probabilities cannot be found by counting outcomes as $\tfrac{1}{2}$ each — they must instead be assigned according to the coin's actual known bias.

Sample-point counting is the skill of correctly finding $n(S)$ (or $n(E)$ for some event $E$) for a given experiment, by one of two methods:

• Direct listing — writing out every sample point explicitly, practical when $S$ is small (e.g. a single coin toss, a single die roll).
• The multiplication principle (introduced in §1) — for an experiment made of several independent stages performed in sequence (coin-and-die together, two dice together, three coins together), multiply the number of outcomes at each stage: $n(S) = n(S_1) \times n(S_2) \times \cdots$. This avoids having to write out a long list by hand and is the standard technique for a compound multi-stage experiment.