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Example:Finding Probability using Bayes Theorem

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Mutually exclusive events


In layman's terms, two events are mutually exclusive if they cannot occur at the same time. An example is tossing a coin once, which can result in either heads or tails, but not both.

In the coin-tossing example, both outcomes are collectively exhaustive, which means that at least one of the outcomes must happen, so these two possibilities together exhaust all the possibilities. However, not all mutually exclusive events are collectively exhaustive. For example, the outcomes 1 and 4 of a single roll of a six-sided die are mutually exclusive (cannot both happen) but not collectively exhaustive (there are other possible outcomes).




Bayes' theorem


In probability theory and applications, Bayes' theorem (alternatively Bayes' law or Bayes' rule) links a conditional probability to its inverse. It is valid in all common interpretations of probability, and is commonly used in science and engineering. The theorem is named for Thomas Bayes (pronounced /ˈbeɪz/ or "bays").


Simple form


Thomas Bayes addressed both the case of discrete probability distributions of data and the more complicated case of continuous probability distributions. In the discrete case, Bayes' theorem relates the conditional and marginal probabilities of events A and B, provided that the probability of B does not equal zero:

where P (α | β) denotes the conditional probability of α given β.


Proportional form

In many situations, P(B) may be thought of as a normalizing constant. If so, Bayes' theorem may be written as


Alternative form

The Law of total probability states that given a partition {Ai}, of the event space,


Bayes' theorem may therefore be more generally written as

The special case of a binary partition may be written as follows. This is useful in the application of Bayesian inference.

(Our solved example in mathguru.com uses this concept)



Bayes' theorem may be extended to take account of more than two events. For example, from the definition of conditional probability and Bayes' theorem:


Similarly, a conditional Bayes' Theorem may be derived:


Derivation using conditional probability

Bayes' theorem may be derived from the definition of conditional probability.





The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.