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The Law of Total Probability

TopThe law of total probability states that, if (An : n = 1, 2, 3, …) is a finite part of a sample space and event An is measurable, then for the other event B of the same probability space,

P(B) = $\sum _n P(B \cap A_n)$
P(B) = $\sum _n P(B | A_n) P_r(A_n)$
Here, for any n for which $P(A_n) = 0$, these terms can be simply omitted from the summation part because $P(B | A_n)$ is finite.

The term law of probability is also known as law of alternatives. The law of total probability theorem is often written as follows:
$P(A) = P(A \cap B) + P(A \cap B')$
Where, P(A) is the probability that the event A occurs. $P(A \cap B)$ refers that the event A and B both occur and $P(A \cap B’)$ is that event A occurs but B doesn’t.

Using the multiple rules, the following expression can be written as follows:
$P(A) = P(A | B) P(B) + P(A | B’) P(B’)$

Baye’s theorem is related to the concept of law of probability. Baye’s theorem provides a new rule with the use of multiplication rule that can be written as follows:
$P(A | B)$ = $\frac{P(A \cap B)}{P(B)}$ = $\frac{P(B | A) P(A)}{P(B)}$

This is the simplest form of Baye’s theorem.

Using the law of total probability, Baye’s theorem can be written as follows:

$P(A | B)$ = $\frac{P(B | A) P(A)}{P(B | A) P(A)}$$ + P(B | A’) P(A’)$

Where, P(A) is the probability that the event A alone occurs.
P(B) is the probability that the event B alone occurs.
P(A’) is the event that A doesn’t occur.
P(A | B) is the probability, given that event A occurs that event B has already occurred.
P(B | A) is the probability that event B occurs given that event A has already occurred.
P(B | A’) is the probability that event B occurs given that event A has already not occurred.

In short, the law of total probability theorem can be written as: if $B_1, B_2, ...., B_n$ is a partition of S such that, $p(B_i)) > 0$ for $i = 1, 2, …., n$, then for any event A,
$P(A) = \sum _{i = 1, 2, ...., n} P(A | B_i) p(B_i)$.