Law of Total Probability

Key idea

Split an event over a partition.

In depth

If B₁, ..., Bₙ partition the sample space, then P(A) = Σ P(A | Bᵢ)P(Bᵢ). Split a problem into cases, find the chance in each, and weigh by how likely each case is.

Worked example

A factory has machine X (60% of output, 2% defects) and Y (40%, 5%): P(defect) = 0.6×0.02 + 0.4×0.05 = 0.032.

Formula

Real-world uses

• Finding the overall defect rate when several factories supply the same parts

Common mistake

Forgetting that the cases must cover all possibilities without overlap.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

Expand all Collapse all

Related expert topics in Probability & Statistics

• Skewness and Kurtosis – Measures of asymmetry and tail heaviness.
• Residuals and R² – Error of a fit and proportion of variance explained.
• Simpson’s Paradox – A trend reverses when groups are combined.
• Bayes’ Theorem – Update beliefs with evidence.
• Medical Tests and Base Rates – Why positive results can still be probably false.
• Linearity of Expectation – Expectation of sums is sum of expectations.