---
title: "Law of Total Probability — AP Stats Definition & Guide"
description: "The law of total probability finds P(A) by summing P(A|B)·P(B) across every branch of a partition. It powers tree diagrams and conditional probability on AP Stats."
canonical: "https://fiveable.me/ap-stats/key-terms/law-of-total-probability"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 2"
---

# Law of Total Probability — AP Stats Definition & Guide

## Definition

The law of total probability says you can find P(A) by splitting the sample space into non-overlapping pieces (a partition), finding the chance of A within each piece, and adding the weighted results: P(A) = P(A|B)·P(B) + P(A|Bᶜ)·P(Bᶜ). It's the math behind every tree diagram in AP Stats.

## What It Is

The law of total probability is a way to compute the [probability](/ap-stats/unit-2/intro-probability/study-guide/gfnBWfyMANOxF3vWLrbA "fv-autolink") of an event when you only know its probability under different conditions. Suppose the [sample space](/ap-stats/key-terms/sample-space "fv-autolink") splits into non-overlapping categories that cover everything (a partition), like "defective machine" vs. "working machine," or "has the disease" vs. "doesn't." Then the overall probability of event A is the sum of each conditional probability times the probability of that condition: P(A) = P(A|B₁)·P(B₁) + P(A|B₂)·P(B₂) + ... 

If that formula looks intimidating, here's the intuitive version. The law of total probability is just a tree diagram written as an equation. Each branch path (condition, then event) gets multiplied, and then you add up every path that ends in A. When an AP problem says "30% of parts come from Factory 1 with a 2% defect [rate](/ap-stats/unit-1/representing-categorical-variable-with-tables/study-guide/JUZVd7cRAnbarZyNoEAg "fv-autolink"), 70% come from Factory 2 with a 5% defect rate, find the probability a random part is defective," you're using this law, even if nobody names it. The answer is (0.30)(0.02) + (0.70)(0.05).

## Why It Matters

This idea lives in [Unit 4](/ap-stats/unit-4 "fv-autolink") (Probability, Random Variables, and Probability Distributions), connected to Topic 4.6 and learning objective 4.6.A, which asks you to calculate probabilities involving combinations of events. The CED doesn't make you memorize the name "law of total probability," but it absolutely tests the skill. Any problem where you're given conditional probabilities for separate groups and asked for an overall probability requires it. It's also the denominator in reverse-conditioning problems (the "given that the test was positive, what's the chance they actually have the disease?" type), which are a classic AP probability setup. If you can build a tree diagram or a [two-way table](/ap-stats/key-terms/two-way-table "fv-autolink") from percentages, you already know this law. You just didn't know it had a name.

## Connections

### [Addition Rule (Unit 4)](/ap-stats/key-terms/addition-rule)

Both rules add probabilities, but the [addition rule](/ap-stats/key-terms/addition-rule "fv-autolink") handles overlapping events, so you subtract P(A ∩ B) to avoid double-counting. The law of total probability adds across a partition, where the pieces can't overlap by design, so nothing gets subtracted.

### [Probability of Union (Unit 4)](/ap-stats/key-terms/probability-of-union)

P(A ∪ B) asks "what's the [chance](/ap-stats/unit-3 "fv-autolink") of A or B?" The law of total probability asks something different: "what's the chance of A, found by routing through B and not-B?" One combines two events; the other decomposes one event into cases.

### Conditional Probability and Tree Diagrams (Unit 4)

The law of total probability is the [multiplication rule](/ap-stats/key-terms/multiplication-rule "fv-autolink"), P(A ∩ B) = P(B)·P(A|B), applied to every branch of a tree and then summed. When you flip a tree around (find P(B|A) from P(A|B)), the law of total probability gives you the denominator.

### Independent Events (Unit 4)

Independence is the special case where the law gets boring. If A is independent of the partition, then P(A|B) = P(A|Bᶜ) = P(A), and the weighted sum just collapses back to P(A). Comparing branch probabilities is actually a quick way to check whether independence holds.

## On the AP Exam

No released FRQ uses the phrase "law of total probability" verbatim, and the AP Stats exam won't ask you to recite the formula by name. Instead, it tests the skill in disguise. Multiple-choice stems give you conditional probabilities for two or three groups (machines, factories, age brackets, test results) plus the size of each group, then ask for an overall probability. Probability FRQs often layer it: part (a) has you find an overall probability using a tree diagram or table (that's the law of total probability), then part (b) reverses the condition and asks for P(group | event), where your part (a) answer becomes the denominator. Your job is to organize the given percentages into a tree or two-way table, multiply along branches, and add the relevant paths, showing your setup clearly for full credit.

## law of total probability vs Addition rule

The addition rule finds the probability of a union: P(A ∪ B) = P(A) + P(B) − P(A ∩ B), and the subtraction exists because A and B might overlap. The law of total probability finds the probability of a single event A by splitting the sample space into mutually exclusive cases and adding P(A|case)·P(case) for each one. There's no overlap correction because a partition has no overlap. Quick test: if the question says "A or B," think addition rule. If it gives you different rates for different groups and asks for one overall rate, think total probability.

## Key Takeaways

- The law of total probability finds P(A) by adding P(A|B)·P(B) across every piece of a partition of the sample space.
- It only works when the conditions form a partition, meaning the categories are mutually exclusive and together cover every possible outcome.
- Every tree diagram calculation where you multiply along branches and add the paths ending in A is the law of total probability in action.
- AP problems rarely name this law; they hand you group sizes and per-group rates and ask for an overall probability.
- It supplies the denominator when you reverse a condition, turning P(A|B) information into a P(B|A) answer.
- Unlike the addition rule, there is no overlap to subtract, because partition pieces can't overlap.

## FAQs

### What is the law of total probability in AP Stats?

It's the rule that the overall probability of an event equals the sum of its conditional probabilities across a partition, weighted by each partition piece's probability: P(A) = P(A|B)·P(B) + P(A|Bᶜ)·P(Bᶜ). On the AP exam it usually shows up as a tree diagram or two-way table problem in Unit 4.

### Do I need to memorize the law of total probability formula for the AP Stats exam?

No, the name and general formula aren't required. But you absolutely need the skill, which the exam tests through tree diagrams and two-way tables under learning objective 4.6.A. If you can multiply along branches and add the paths, you've got it.

### How is the law of total probability different from the addition rule?

The addition rule finds P(A or B) for two possibly overlapping events, so you subtract P(A ∩ B). The law of total probability finds P(A) alone by splitting the sample space into non-overlapping cases, so there's nothing to subtract. Different question, different tool.

### Is the law of total probability the same as a tree diagram?

Essentially yes. A tree diagram is the law of total probability drawn as a picture. Multiplying along each branch gives P(A|B)·P(B), and adding the paths that end in your event gives the total probability.

### When do I use the law of total probability on an FRQ?

Use it whenever a problem gives you different rates for different groups, like 30% of items from one source with a 2% defect rate and 70% from another with 5%, and asks for one overall probability. You'd compute (0.30)(0.02) + (0.70)(0.05) = 0.041. It's also step one before answering any "given the result, which group?" follow-up.

## Related Study Guides

- [2.7 Independent Events and Unions of Events](/ap-stats/unit-2/independent-events-unions-events/study-guide/aMOuOhtDQIJAtgZx3JNz)

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