Enter the conditional probability of event A given each event Bi and the probability of each event Bi into the calculator to determine the total probability of event A.

Law Of Total Probability Formula

The following formula is used to calculate the probability of an event based on the Law of Total Probability. 

P(A) = Σ [P(A | Bi) * P(Bi)]

Variables:

  • P(A) is the total probability of event A
  • P(A | Bi) is the conditional probability of event A given event Bi
  • P(Bi) is the probability of event Bi
  • The summation Σ is over all possible events Bi in the partition of the sample space

To calculate the total probability of an event A, sum up the products of the conditional probability of event A given each event Bi and the probability of each event Bi. This is done for all possible events Bi 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.

How to Calculate Law Of Total Probability?

The following steps outline how to calculate the Law of Total Probability.

  1. First, identify the events Bi that form a partition of the sample space.
  2. Next, determine the probability of each event Bi, denoted as P(Bi).
  3. Next, determine the conditional probability of event A given each event Bi, denoted as P(A | Bi).
  4. Next, multiply each conditional probability P(A | Bi) by the probability of the corresponding event Bi, P(Bi).
  5. Finally, sum up all the products obtained in the previous step to calculate the total probability of event A, denoted as P(A).

Example Problem : 

Use the following variables as an example problem to test your knowledge.

P(A) = ?

P(A | B1) = 0.3

P(A | B2) = 0.6

P(B1) = 0.4

P(B2) = 0.6