Law of Total Probability

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Calculate

Example: a factory has three machines that make 50%, 30% and 20% of its parts, with defect rates of 1%, 2% and 3%. What is the chance that a part picked at random is defective? Change the values to work out your own case.

Probability of each scenario P(A_i) (%) The scenarios must cover every case without overlapping: they add up to 100%. Probability of B in each scenario P(B|A_i) (%) In the same order as the scenarios above.

Step by step

What each term means

Total Probability of B: The overall probability of event B, taking into account every possible scenario or partition of the sample space.
Summation: Adds over all the partitions A₁, A₂, ..., Aₙ, which together form a complete and mutually exclusive partition of the sample space.
Conditional Probability P(B|Aᵢ): The probability of B given that Aᵢ happened. It is the probability of B within each specific scenario Aᵢ.
Probability of Aᵢ: The probability of each partition Aᵢ. The events Aᵢ must be mutually exclusive and exhaustive (their union covers the whole sample space).