List of formulas related to total probability and Bayes’ formulas:

Formula of total probability

If (B_1, B_2, \ldots, B_n) is a complete system of events, then the formula for the total probability looks like this:

[
P(A) = \sum_{i=1}^{n} P(A | B_i) P(B_i)
]

Where:

  • ( P(A) ) — probability of event ( A ),
  • ( P(A | B_i) ) — the conditional probability of an event ( A ) given that an event ( B_i ) has occurred,
  • ( P(B_i) ) — probability of event ( B_i ).

Bayes’ formula

Bayes’ formula allows us to calculate the conditional probability of an event (A) given an event (B):

[
P(A | B) = \frac{P(B | A) P(A)}{P(B)}
]

Where:

  • ( P(A | B) ) — the conditional probability of an event ( A ) given that an event ( B ) has occurred,
  • ( P(B | A) ) — the conditional probability of an event ( B ) given that an event ( A ) has occurred,
  • ( P(A) ) — probability of event ( A ),
  • ( P(B) ) — probability of event ( B ).

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