Total probability theorem Calculator

Formula first

Overview

This theorem allows for the calculation of the probability of an event B by considering all possible mutually exclusive and exhaustive scenarios (A and its complement $A^c$) that could lead to B. It is a fundamental tool in engineering for breaking down complex probabilistic systems into manageable conditional components. By summing the weighted probabilities of these scenarios, one can determine the overall likelihood of the event occurring.

Symbols

Variables

$\mathbf{P}$(B) = Total Probability P(B), $\mathbf{P}$(A) = Probability of A, $\mathbf{P}$(B$|$ A) = Conditional Probability P(B|A), $\mathbf{P}$($A^c$) = Probability of A complement, $\mathbf{P}$(B$|$ $A^c$) = Conditional Probability P(B|$A^c$)

Apply it well

When To Use

When to use: Use this theorem when you need to find the probability of an event B, but you only have information about the conditional probabilities of B given some event A and its complement A^c.

Why it matters: It is essential for Bayesian inference, reliability engineering, and decision analysis where outcomes depend on underlying, often hidden, states.

Avoid these traps

Common Mistakes

• Failing to ensure that A and $A^c$ are exhaustive, meaning they do not cover all possible outcomes.
• Confusing P(B|A) with P(A|B), which requires Bayes' Theorem.

One free problem

Practice Problem

A factory uses two machines to produce parts. Machine A produces 60% of the parts, and Machine B produces 40%. If 5% of parts from Machine A are defective and 10% of parts from Machine B are defective, what is the total probability that a randomly selected part is defective?

Solve for: $P_B$

Hint: Use the formula P(B) = P(A)P(B|A) + P($A^c$)P(B|$A^c$).

The full worked solution stays in the interactive walkthrough.

References

Sources

1. Ross, S. M. (2019). A First Course in Probability (10th ed.). Pearson.
2. Papoulis, A., & Pillai, S. U. (2002). Probability, Random Variables and Stochastic Processes (4th ed.). McGraw-Hill.
3. Wikipedia, Total probability theorem
4. Wolfram MathWorld, Law of Total Probability