Law of Total Probability Calculator

Formula

The formula to calculate the total probability of event A (P(A)) is:

\[ P(A) = \sum [P(A | B_i) \times P(B_i)] \]

Where:

\(P(A)\) is the total probability of event A
\(P(A | B_i)\) is the conditional probability of event A given event \(B_i\)
\(P(B_i)\) is the probability of event \(B_i\)
The summation \(\sum\) is over all possible events \(B_i\) in the partition of the sample space

What is the Law of Total Probability?

The Law of Total Probability is a fundamental rule in statistics that provides a method to calculate the probability of an event based on its occurrence under several distinct conditions or partitions. It states that the total probability of an outcome is the sum of the probabilities of that outcome occurring in each of the mutually exclusive scenarios that cover all possible outcomes. This law is particularly useful when the probabilities of the event under each condition are easier to compute than the probability of the event directly.

Example Calculation

Let's assume the following values:

P(A | B1) = 0.3, P(B1) = 0.5
P(A | B2) = 0.6, P(B2) = 0.3
P(A | B3) = 0.2, P(B3) = 0.2

Using the formula to calculate the total probability of event A (P(A)):

\[ P(A) = (0.3 \times 0.5) + (0.6 \times 0.3) + (0.2 \times 0.2) = 0.15 + 0.18 + 0.04 = 0.37 \]

The total probability of event A is 0.37.