Statistics - Probability Bayes Theorem

One of the most significant developments in the probability field has been the development of Bayesian decision theory which has proved to be of immense help in making decisions under uncertain conditions. The Bayes Theorem was developed by a British Mathematician Rev. Thomas Bayes. The probability given under Bayes theorem is also known by the name of inverse probability, posterior probability or revised probability. This theorem finds the probability of an event by considering the given sample information; hence the name posterior probability. The bayes theorem is based on the formula of conditional probability.

conditional probability of event ${A_1}$ given event ${B}$ is

${P(A_1/B) = \frac{P(A_1\ and\ B)}{P(B)}}$

Similarly probability of event ${A_1}$ given event ${B}$ is

${P(A_2/B) = \frac{P(A_2\ and\ B)}{P(B)}}$

Where

${P(B) = P(A_1\ and\ B) + P(A_2\ and\ B) \\[7pt] P(B) = P(A_1) \times P (B/A_1) + P (A_2) \times P (BA_2) }$

${P(A_1/B)}$ can be rewritten as

${P(A_1/B) = \frac{P(A_1) \times P (B/A_1)}{P(A_1)} \times P (B/A_1) + P (A_2) \times P (BA_2)}$

Hence the general form of Bayes Theorem is

${P(A_i/B) = \frac{P(A_i) \times P (B/A_i)}{\sum_{i=1}^k P(A_i) \times P (B/A_i)}}$

Where ${A_1}$, ${A_2}$...${A_i}$...${A_n}$ are set of n mutually exclusive and exhaustive events.

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