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Probability Bayes Theorem
Statistics - Probabipty Bayes Theorem
One of the most significant developments in the probabipty 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 probabipty given under Bayes theorem is also known by the name of inverse probabipty, posterior probabipty or revised probabipty. This theorem finds the probabipty of an event by considering the given sample information; hence the name posterior probabipty. The bayes theorem is based on the formula of conditional probabipty.
conditional probabipty of event ${A_1}$ given event ${B}$ is
${P(A_1/B) = frac{P(A_1 and B)}{P(B)}}$Similarly probabipty 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) imes P (B/A_1) + P (A_2) imes P (BA_2) }$ ${P(A_1/B)}$ can be rewritten as ${P(A_1/B) = frac{P(A_1) imes P (B/A_1)}{P(A_1)} imes P (B/A_1) + P (A_2) imes P (BA_2)}$Hence the general form of Bayes Theorem is
${P(A_i/B) = frac{P(A_i) imes P (B/A_i)}{sum_{i=1}^k P(A_i) imes P (B/A_i)}}$Where ${A_1}$, ${A_2}$...${A_i}$...${A_n}$ are set of n mutually exclusive and exhaustive events.
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