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Bayes formula and examples of tasks

Lecture



Let event B occur simultaneously with one of the n incompatible events A1, A2, A3, ... An . It is required to find the probability of event Ai , if it is known that event B has occurred.

Based on the probability theorem of the product of two events, one can write

Bayes formula and examples of tasks

From where

Bayes formula and examples of tasks

or

Bayes formula and examples of tasks (3.2)

Formula (3.2) is called the Bayes formula .

Example. Three organizations submitted accounts to the control department for random inspection. The first organization submitted 15 invoices, the second - 10, the third - 25. The probabilities of the correct issuance of invoices for these organizations are known and, accordingly, are: 0.9; 0.8; 0.85. One account was selected and it turned out to be correct. Determine the probability that this account belongs to the second organization.

Decision. Let be Bayes formula and examples of tasks - Account selection events at the first, second and third organizations. The corresponding probabilities will be

Bayes formula and examples of tasks , Bayes formula and examples of tasks , Bayes formula and examples of tasks

According to the formula of total probability, we determine the probability of choosing a properly executed account.

Bayes formula and examples of tasks

By the Bayesian formula we find the initial probability.

Bayes formula and examples of tasks .


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis