## Probability

# Bayes' Theorem

**Law of Total Probability:**- Let S be the sample space and let E
_{1}, E_{2}, E_{3}, ....... En are the n-mutually exclusive and exhaustive events associated with the random experiment. Let A be simple event associated with one of these elementary events. Then \tt P(A) = P(E_{1})P(A(E_{1})) + P(E_{2})P(A(E_{2})) + ....... + P(E_{n})P(A(E_{n})) **Baye's Theorem**- Let S be the sample space and let E
_{1}, E_{2}, E_{3}..... E_{n}are the n-mutually exclusive and exhaustive events associated with the random experiment. Let A be the simple event associated with one of these elementary events. Then

\tt P(E_{K}/A) = \frac{P(A/E_{K})P(E_{K})}{P(E_{1})P(A/E_{1}) + P(E_{2})P(A/E_{2}) + ..... P(E_{K})P(A/E_{K}) + ...... + P(E_{n})P(A/E_{n})}

### View the Topic in this video From 15:06 To 43:30

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1. If A is any event which occurs with E_{1} or E_{2} or ..... or E_{n}, then P(A) = P(E_{1}) P(A / E_{1}) + P(E_{2}) P(A / E_{2}) + .... + P(E_{n}) P(A / E_{n})

=\sum_{r=1}^n P(E_{r})P(A / E_{r})

2. If A is any event which occurs with E_{1} or E_{2} or ..... or E_{n}, then probability of occurrence of E_{i}, when A occurred,

P(E_{i} / A)=\frac{P(E_{i})P(A/E_{i})}{\sum_{i=1}^{n} P(E_{i})P(A/E_{i})}, i = 1, 2, ... , n