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Sum rule of probability

Sum rule of probability describes the probability of the combination of two events (A or B) (A \lor B).
In set theory symbols this is equivalent to the event C the union of events A and B:
C = A \cup B

The following special cases can be distinguished.

Mutually exclusive events

Two sets of events are mutually exclusive if the intersection between these two sets is empty.

A \cap B = \emptyset

For two mutually exclusive events the probabilities can be added.

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

Identical events

Two sets of events are identical if the intersection and union of these tow sets are identical.

A \cap B = A \cup B = A = B

For two identical events the probability is equal to each of the events.

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

Not mutually exclusive events

This is the general case from which all the special cases above can be deduces.

A \cap B \ne \emptyset
A \ne B

For not mutually exclusive events we can write:

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

using the multiplication rule this can be transformed into

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

General rule

Generally the addition rule can be given for multiple events.

P(\bigcup_{i=1}^n A_i) = \sum_{i=1}^n P(A_i) - \sum_{i,j:i<j} P(A_i \cap A_j) + \sum_{i,j,k:i<j<k} P(A_i \cap A_j \cap A_k) - \cdot \cdot \cdot +(-1)^{n-1} P(\bigcap_{i=1}^n A_i)

More compactly written.

P\left(\bigcup_{i=1}^n A_i\right) = \sum_{k=1}^n (-1)^{k+1} \left(\sum_{1 \le i_{1} < \cdot \cdot \cdot < i_{k} \le n} P(A_{i_{1}} \cap \cdot \cdot \cdot \cap A_{i_{k}}) \right)

For proof of this rule see the wiki page.

Tags: Statistics

Categories: Mathematics


(c) Mato Nagel, Weißwasser 2004-2013, Disclaimer