If we assemble a deck of 52
playing cards with no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as each card is a possible outcome. An event, however, is any subset of the sample space, including any
singleton set (an
elementary event), the
empty set (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one). Other events are
proper subsets of the sample space that contain multiple elements. So, for example, potential events include: of an event. B is the sample space and A is an event.By the ratio of their areas, the probability of A is approximately 0.4. • "Red and black at the same time without being a joker" (0 elements), • "The 5 of Hearts" (1 element), • "A King" (4 elements), • "A Face card" (12 elements), • "A Spade" (13 elements), • "A Face card or a red suit" (32 elements), • "A card" (52 elements). Since all events are sets, they are usually written as sets (for example, {1, 2, 3}), and represented graphically using
Venn diagrams. In the situation where each outcome in the sample space Ω is equally likely, the probability P of an event A is the following : \mathrm{P}(A) = \frac\,\ \left( \text{alternatively:}\ \Pr(A) = \frac\right) This rule can readily be applied to each of the example events above. ==Events in probability spaces== Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard
probability distributions, such as the
normal distribution, the sample space is the set of real numbers or some subset of the
real numbers. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers
'badly behaved' sets, such as those that are
nonmeasurable. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as
joint and
conditional probabilities, to work, it is necessary to use a
σ-algebra, that is, a family closed under complementation and countable unions of its members. The most natural choice of
σ-algebra is the
Borel measurable set derived from unions and intersections of intervals. However, the larger class of
Lebesgue measurable sets proves more useful in practice. In the general
measure-theoretic description of
probability spaces, an event may be defined as an element of a selected
-algebra of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the -algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, all are elements of the -algebra. ==A note on notation==