Sample space

The sample space, also called the space of possibilities, denotes the set of all the results a random experiment can produce. It is the frame of reference within which all probabilistic reasoning takes place: before measuring the chance of anything, one must first know which results are conceivable.

The mental image is that of a complete inventory. One draws up an exhaustive list of the outcomes, without forgetting any or counting any twice. For a six-sided die, the sample space is the set {1, 2, 3, 4, 5, 6}; for a coin, it is {heads, tails}; for drawing a card, it has 52 possible results.

The number of elements in the sample space plays a direct role in the calculation. When the outcomes are equally likely, the probability of an event is obtained by dividing the number of favorable outcomes by the total number of outcomes in the sample space. A sample space of 6 outcomes gives each result a base probability of 1 in 6.

An event is always a subset of the sample space: it groups certain outcomes among all those listed. The whole sample space itself corresponds to the certain event, with probability 1, and the empty set to the impossible event, with probability 0.

The most common trap is to define the sample space poorly, forgetting outcomes or describing them at an unsuitable level of detail. For two coins flipped together, the correct sample space has four distinct outcomes, heads-heads, heads-tails, tails-heads, and tails-tails, and not three; neglecting this distinction immediately distorts the probabilities.

Each tool on the site implicitly rests on a sample space. The wheel of fortune has its sectors as its sample space, the number generator the chosen range, the name draw the list of participants: properly identifying this set is the first step toward correctly interpreting a result.