section_path: Mathematical treatment
section_name: Mathematical treatment
paragraph_text: Mathematical treatment
  
  Consider an experiment that can produce a number of results. The collection of all possible results is called the sample space of the experiment, sometimes denoted as Ω {\displaystyle \Omega }. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling a die can produce six possible results. One collection of possible results gives an odd number on the die. Thus, the subset {1,3,5} is an element of the power set of the sample space of dice rolls. These collections are called "events". In this case, {1,3,5} is the event that the die falls on some odd number. If the results that actually occur fall in a given event, the event is said to have occurred.
  
  A probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) is assigned a value of one. To qualify as a probability, the assignment of values must satisfy the requirement that for any collection of mutually exclusive events (events with no common results, such as the events {1,6}, {3}, and {2,4}), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events.
  
  The probability of an event A is written as P (A) {\displaystyle P(A)}, p (A) {\displaystyle p(A)}, or Pr (A) {\displaystyle {\text{Pr}}(A)}. This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using the concept of a measure.
  
  The opposite or complement of an event A is the event (that is, the event of A not occurring), often denoted as A ′, A c {\displaystyle A',A^{c}}, A ¯, A ∁, ¬ A {\displaystyle {\overline {A}},A^{\complement },\neg A}, or ∼ A {\displaystyle {\sim }A}; its probability is given by P (not A) = 1 − P (A). As an example, the chance of not rolling a six on a six-sided die is 1 – (chance of rolling a six) = 1 − 1 / 6 = 5 / 6. For a more comprehensive treatment, see Complementary event.
  
  If two events A and B occur on a single performance of an experiment, this is called the intersection or joint probability of A and B, denoted as P (A ∩ B). {\displaystyle P(A\cap B).}