Explanation:
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).
http://en.wikipedia.org/wiki/Mutually_exclusive_events
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 {A_{i}}, 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)
Extensions
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.
http://en.wikipedia.org/wiki/Bayes'_theorem
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