Law of total probability
Combine conditional probabilities across every mutually exclusive case.
Law of total probability is one of 2 events formulas in the probability section of this library, and it is used at ap · university level.
Why law of total probability works
Split the sample space into cases that do not overlap and leave nothing out. Every way the event can occur sits inside exactly one case, so adding up the chance within each case, weighted by how likely that case is, rebuilds the whole probability without missing or repeating anything.
What each symbol means
The events $A_i$ form a partition of the sample space.
Law of total probability: when it holds
The $A_i$ must be disjoint, exhaustive, and have positive probability when conditioned upon.
When it stops applying
The cases must be mutually exclusive and cover everything. If your categories overlap, such as people who take the bus and people who take the train when some take both, the weighted sum double counts those people and can produce an impossible answer above 1.
Law of total probability: a worked example
If two groups have shares $.4,.6$ and event rates $.2,.5$, then $P(B)=.4(.2)+.6(.5)=.38$.
The mistake to avoid
What people do: Adding the conditional probabilities without weighting them by the case probabilities.
Why it goes wrong: With group shares of 0.4 and 0.6 and event rates of 0.2 and 0.5, adding the rates gives 0.7. The weighted total is 0.38, and the unweighted sum can even climb past 1 with enough cases.
Do this instead: Multiply each conditional probability by the probability of its own case before adding, and check that the case probabilities themselves add to 1.
Law of total probability: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. The events $A_i$ form a partition of the sample space.
- Check the conditions before substituting. The $A_i$ must be disjoint, exhaustive, and have positive probability when conditioned upon.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most probability slips.
Where this formula fits
- Subject
- Probability formulas — 17 entries in this library
- Topic
- Events
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where law of total probability comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study probability — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 probability formulas — the full grouped reference, or the complete formula library.
Questions about law of total probability
How do I check that my split is a genuine partition?
Confirm the case probabilities add to exactly 1 and that no outcome belongs to two cases. Both conditions have to hold, since covering everything is not the same as not overlapping.
Can I use more than two cases?
Yes, any number, including infinitely many as long as the total probability still comes to 1. Each case simply adds another weighted term to the sum.
How does this relate to a tree diagram?
Each branch of the tree is one case, multiplying along a branch gives one term, and adding the branches that end in the event gives this formula exactly.
Where does this appear inside Bayes' theorem?
It supplies the denominator. When the overall chance of the evidence is not given directly, this is how you assemble it from the individual cases.