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 ).