Impossible event
Encyclopedia
In the mathematics of probability
, an impossible event is an event
A with probability
zero, or Pr(A) = 0. See in particular almost surely
.
An impossible event is not the same as the stronger concept of logical impossibility. For any continuous probability distribution the probability of any single elementary event
is 0, yet the event is not logically impossible as an event outside the distribution. For instance, the probability of hitting any specific point on a dart board, let's say a square in Cartesian coordinates [−10, 10] × [−10, 10] and the point (4.5678, −8.4568), is 0, because there is an uncountably infinite
number of points on the board. In contrast, hitting a point outside of the space considered is logically impossible.
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
, an impossible event is an event
Event (probability theory)
In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...
A with probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
zero, or Pr(A) = 0. See in particular almost surely
Almost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
.
An impossible event is not the same as the stronger concept of logical impossibility. For any continuous probability distribution the probability of any single elementary event
Elementary event
In probability theory, an elementary event or atomic event is a singleton of a sample space. An outcome is an element of a sample space. An elementary event is a set containing exactly one outcome, not the outcome itself...
is 0, yet the event is not logically impossible as an event outside the distribution. For instance, the probability of hitting any specific point on a dart board, let's say a square in Cartesian coordinates [−10, 10] × [−10, 10] and the point (4.5678, −8.4568), is 0, because there is an uncountably infinite
Uncountable set
In mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.-Characterizations:There...
number of points on the board. In contrast, hitting a point outside of the space considered is logically impossible.