The law of total probability helps you find the probability of an event by breaking the sample space into separate cases. It is useful when an outcome can happen through several different paths, such as choosing from different groups, machines, classes, or weather conditions. Instead of trying to count event A all at once, you split the problem into partitions B1, B2, B3, and so on.
This makes complex probability questions easier to organize and solve.
Key Facts
- If B1, B2, ..., Bn form a partition of the sample space, then P(A) = P(A|B1)P(B1) + P(A|B2)P(B2) + ... + P(A|Bn)P(Bn).
- A partition means the events B1, B2, ..., Bn are mutually exclusive and together cover the whole sample space S.
- Mutually exclusive means P(Bi and Bj) = 0 for i not equal to j.
- The multiplication rule gives P(A and Bi) = P(A|Bi)P(Bi).
- The law of total probability can also be written as P(A) = sum from i = 1 to n of P(A and Bi).
- Bayes' theorem uses total probability in the denominator: P(Bk|A) = P(A|Bk)P(Bk) / P(A).
Vocabulary
- Sample space
- The sample space is the set of all possible outcomes of a random experiment.
- Partition
- A partition is a collection of non-overlapping events that together include every outcome in the sample space.
- Conditional probability
- Conditional probability P(A|B) is the probability that event A occurs given that event B has occurred.
- Mutually exclusive
- Mutually exclusive events cannot happen at the same time.
- Bayes' theorem
- Bayes' theorem updates the probability of a cause or condition after observing new evidence.
Common Mistakes to Avoid
- Forgetting to check that the B events form a partition is wrong because the law of total probability requires the cases to be non-overlapping and complete.
- Adding P(A|B1), P(A|B2), and P(A|B3) directly is wrong because each conditional probability must be weighted by the probability of its case.
- Using P(A|B) as if it were the same as P(B|A) is wrong because reversing the condition usually changes the probability.
- Leaving out one possible case is wrong because the total probability must include every way event A can occur within the sample space.
Practice Questions
- 1 A factory uses Machine 1 for 60% of its products and Machine 2 for 40%. Machine 1 makes defective products 2% of the time, and Machine 2 makes defective products 5% of the time. What is the probability that a randomly selected product is defective?
- 2 A school has 30% freshmen, 25% sophomores, 25% juniors, and 20% seniors. The probabilities that a student in each group takes physics are 0.40, 0.35, 0.30, and 0.25 respectively. What is the probability that a randomly selected student takes physics?
- 3 Explain why the law of total probability would not be valid if B1, B2, B3, and B4 overlap or fail to cover the entire sample space.