Law of total probability

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In mathematics, the law of total probability states that if A and B are events, then

P(B) = P(B|A)P(A) + P\left(B|A^c\right)P\left(A^c\right)

(see [1])

In general, if A_1, \ldots, A_n are mutually exclusive events that partition the sample space, and B is any event, then

P(B) = \sum_{i=1}^n P\left(B|A_i\right) P\left(A_i\right)

(see [2])


  1. Ghahramani, Saeed. Fundamentals of Probability: With Stochastic Processes. 3rd ed. Upper Saddle River (NJ): Pearson Prentice Hall, 2005. p. 88
  2. Ghahramani, p. 93
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