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Law of Total Probability Formula:
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The Law of Total Probability is a fundamental rule in probability theory that relates marginal probabilities to conditional probabilities. It allows us to compute the probability of an event A by considering all possible ways that A can occur through a partition of the sample space.
The calculator uses the Law of Total Probability formula:
Where:
Explanation: The formula breaks down the probability of event A into cases based on whether event B occurs or not, then combines these cases using the rules of probability.
Details: This law is widely used in statistics, machine learning, risk assessment, and decision analysis. It's particularly useful when dealing with hierarchical or conditional probability models and when we need to find probabilities of events that can occur through multiple pathways.
Tips: Enter all three required probability values between 0 and 1. The calculator will automatically compute P(¬B) as 1 - P(B) and then calculate P(A) using the law of total probability.
Q1: Can this formula be extended to more than two partitions?
A: Yes, the law can be generalized to any partition of the sample space: P(A) = Σ P(A|B_i) × P(B_i) for a complete set of mutually exclusive events B_i.
Q2: What if I have more than two conditioning events?
A: You would need to use the generalized form of the law that sums over all possible conditioning events in your partition.
Q3: Are there any restrictions on using this formula?
A: The events B and ¬B must form a partition of the sample space (they must be mutually exclusive and exhaustive).
Q4: How is this related to Bayes' Theorem?
A: The Law of Total Probability is often used in the denominator of Bayes' Theorem to compute prior probabilities.
Q5: Can I use percentages instead of probabilities?
A: Yes, but you would need to convert percentages to probabilities (divide by 100) before using this calculator.