probability

(NL: kans, waarschijnlijkheid)

A probability is a number that indicates how likely it is that a certain event will happen during an experiment. In other words, the probability represents the relative frequency of the occurrence of the event at hand when performing a large number of (independent) experiments.

universe, sample space

The set of all possible outcomes of an event is called the sample space or universe, noted as \(\Omega\) (Greek capital letter Omega).

Events are subsets of $\Omega $ and are typically noted as a capital letter (e.g. \(A, B, \ldots\)).

axioms of probability

Probabilities must meet the following conditions:

1. Probabilities are nonnegative: \(P(A) \geq 0\) for each \(A\)
2. The universe has probability 1: \(P(\Omega) = 1\)

3. Sum rule: if \(A\) and \(B\) are disjuoint events (i.e. \(A \cap B = \emptyset\)), then it holds that:

   \(P(A\cup B) = P(A) + P(B).\)

properties

From the three axioms of probability, we can derive \emph{all}\/ properties of probabilities.

Some important ones are listed below:

• Complement rule: For each event \(A\), it holds that:

  \(P(\overline{A}) = 1 - P(A)\)

  with \(\overline{A} = \Omega \setminus A\), the probability that event \(A\) does NOT occur.

• The impossible event has probability zero: \(P(\emptyset) = 0\).

• The general sum rule: For each event \(A\) and \(B\), it holds that:

  \(P(A\cup B) = P(A) + P(B) - P(A\cap B)\)

independent events

Two events are independent if and only if:

\(P(A \cap B) = P(A)P(B)\)

contitional probability

The probability that event \(A\) will occur on the condition that event \(b\) will occur, is notated with \(P(A|B)\)