Source: STAT 350 Practice Exam 1, Purdue University
Tags: probability, independence, mutually exclusive, complement, inclusion-exclusion, conditional probability, addition rule, sample space, events
Difficulty: Foundational Prerequisites: Basic set theory notation (unions, intersections, complements). Comfort with the idea of a sample space.
Probability rules are the grammar of statistics. Every later topic in STAT 350, from distributions to hypothesis testing, rests on being able to manipulate events, compute unions and intersections, and apply conditional probability correctly. If you are coming in cold, this is the place to start: the rules here are small in number but show up everywhere, and confusing independence with mutual exclusivity is one of the most common mistakes on any introductory statistics exam. You should already be comfortable with basic set notation and the idea that probabilities are numbers between 0 and 1.
Probability rules govern how we combine and relate events. The inclusion-exclusion formula, the complement rule, and the definition of conditional probability are always true and form the backbone of every probability calculation. Independence and mutual exclusivity are different properties that are often confused but almost never overlap.
Sample space (Omega)
The set of all possible outcomes of an experiment. Every event is a subset of Omega. In simple terms, this is the "universe" of things that could happen.
Event
A subset of the sample space. An event "occurs" when the outcome of the experiment falls inside that subset. Think of it as a specific collection of outcomes you care about.
P(A), probability of event A
A number between 0 and 1 (inclusive) representing the likelihood of event A occurring. In simple terms, it is a measure of how likely A is.
Intersection (A and B)
Written A ∩ B. The event that both A and B occur. Think of it as "A AND B both happen."
Union (A or B)
Written A ∪ B. The event that at least one of A or B occurs. Think of it as "A OR B (or both) happens."
Complement
Written A' or A^c. The event that A does not occur. P(A') = 1 - P(A). In simple terms, everything in the sample space that is not in A.
Mutually exclusive (disjoint)
Two events A and B such that A ∩ B = empty set, meaning they cannot both occur at the same time. P(A ∩ B) = 0. Think of it as: if one happens, the other cannot.
Independent events
Two events A and B such that knowing one occurred does not change the probability of the other. Formally: P(A ∩ B) = P(A) · P(B), or equivalently P(A|B) = P(A). In simple terms, one event's occurrence has no effect on the other.
Conditional probability
P(A|B) = P(A ∩ B) / P(B), defined when P(B) > 0. The probability of A given that B has occurred. Think of it as "zooming in" on the world where B happened and asking how likely A is within that world.
Inclusion-exclusion formula (addition rule)
P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Always true. In simple terms, add the two probabilities but subtract the overlap so you do not double-count it.
P(A ∩ B) = P(A|B) · P(B). This is simply a rearrangement of the definition of conditional probability and holds for all events where P(B) > 0.
Equivalently, P(A ∩ B) = P(B|A) · P(A).
This is not a test for independence. The equation P(A ∩ B) = P(A|B) · P(B) is always true regardless of whether A and B are independent.
For independent events specifically, P(A|B) = P(A), which simplifies the multiplication rule to P(A ∩ B) = P(A) · P(B).
The fact that P(A ∩ B) = P(A|B) · P(B) tells you nothing about whether events are independent. You would need to check whether P(A|B) = P(A), or equivalently whether P(A ∩ B) = P(A) · P(B).
P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Always true.
Rearranging: P(A ∩ B) = P(A) + P(B) - P(A ∪ B). Also always true.
The simplified version P(A ∪ B) = P(A) + P(B) only holds when A and B are mutually exclusive (because P(A ∩ B) = 0).
Be careful: P(A ∪ B) = P(A) - P(A ∩ B) is not generally true. This would imply P(B) = 0, which contradicts P(B) > 0.
Mutually exclusive: P(A ∩ B) = 0. The events cannot occur together.
Complementary: A and B are complements if they are mutually exclusive AND P(A ∪ B) = 1. This means A' = B.
Mutually exclusive events are not necessarily complements. There can be outcomes in neither A nor B, so P(A) + P(B) can be less than 1.
The statement "if A and B are mutually exclusive, then P(A) = 1 - P(B)" is only true if A and B are also exhaustive (cover the whole sample space), making them complements.
P(A') = 1 - P(A).
P(A' ∩ B') = P((A ∪ B)') = 1 - P(A ∪ B). This is De Morgan's law applied with the complement rule. "Neither A nor B" is the complement of "A or B."
This is extremely useful for "failed both" or "none of the above" type problems.
Conditional probability: P(A|B) = P(A ∩ B) / P(B)
Multiplication rule: P(A ∩ B) = P(A|B) · P(B)
Addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Complement: P(A') = 1 - P(A)
De Morgan's: P(A' ∩ B') = 1 - P(A ∪ B)
Independence test: P(A ∩ B) = P(A) · P(B)
Insurance companies use these exact rules when pricing policies that cover multiple types of risk. The probability that a home suffers either fire damage or water damage in a given year uses the inclusion-exclusion formula, and the question of whether those two risks are independent determines how joint coverage is priced. The conditional probability formula is the basis of every medical diagnostic test: "given a positive result, what is the probability the patient has the disease?"
Students often think that P(A ∩ B) = P(A|B) · P(B) being true means events are independent. It does not. This equation is the definition of conditional probability and holds for all events.
Students frequently confuse mutually exclusive with independent. They are almost opposite: if A and B are mutually exclusive with P(A) > 0 and P(B) > 0, they cannot be independent, because knowing A occurred tells you B did not.
Students sometimes assume mutually exclusive events are complements. Complements require both mutual exclusivity and exhaustiveness (they must cover the entire sample space).
When computing "the probability of neither," students sometimes try to multiply individual complement probabilities. Use De Morgan's law instead: P(A' ∩ B') = 1 - P(A ∪ B).
⚠️ The multiplication rule P(A ∩ B) = P(A|B)P(B) is always true. Independence requires the stronger condition P(A ∩ B) = P(A)P(B).
⚠️ "Mutually exclusive" does not mean "complementary." P(A) = 1 - P(B) requires A and B to be complements, not merely disjoint.
⚠️ The inclusion-exclusion rearrangement P(A ∩ B) = P(A) + P(B) - P(A ∪ B) is always true and is tested frequently.
⚠️ For "failed both" problems, use the complement of the union: P(A' ∩ B') = 1 - P(A ∪ B).
True or false: P(A ∩ B) = P(A|B) · P(B) is only true when A and B are independent.
True or false: If A and B are mutually exclusive and P(A) > 0, P(B) > 0, then A and B can also be independent.
Fill in the blank: P(A' ∩ B') = 1 - ________.
True or false: If P(A) = 0.4 and P(B) = 0.5 and A, B are mutually exclusive, then P(A) = 1 - P(B).
Fill in the blank: P(A ∩ B) = P(A) + P(B) - ________.
Answers: 1. False (it is always true). 2. False (they cannot be independent if both have positive probability). 3. P(A ∪ B). 4. False (P(A) + P(B) = 0.9, not 1, so they are not complements). 5. P(A ∪ B).
Q: Let A and B be events in the same sample space with P(A) > 0 and P(B) > 0. If P(A ∩ B) = P(A|B)P(B), must A and B be independent?
A: No. The equation P(A ∩ B) = P(A|B)P(B) is the definition of conditional probability and is always true. Independence requires the additional condition that P(A|B) = P(A), which simplifies this to P(A ∩ B) = P(A)P(B). The given equation alone tells us nothing about independence.
Q: Let A and B be events with P(A) > 0 and P(B) > 0. If A and B are mutually exclusive, must it be true that P(A) = 1 - P(B)?
A: No. P(A) = 1 - P(B) means P(A) + P(B) = 1. Mutually exclusive only guarantees P(A ∩ B) = 0, meaning P(A ∪ B) = P(A) + P(B). But P(A ∪ B) could be less than 1 (there may be outcomes in neither A nor B), so P(A) + P(B) is not necessarily 1.
Q: Which of the following is always true? (a) P(A ∪ B) = P(A) + P(B), (b) P(A ∪ B) = P(A) - P(A ∩ B), (c) P(A ∩ B) = P(A) + P(B) - P(A ∪ B), (d) more than one, (e) none.
A: (c). This is a rearrangement of the inclusion-exclusion formula. Option (a) requires mutual exclusivity. Option (b) would require P(B) = 0.
Q: An exam has a written component and an oral component. 75% passed written, 85% passed oral, and 95% passed at least one. What percentage failed both?
A: P(failed both) = P(W' ∩ O') = 1 - P(W ∪ O) = 1 - 0.95 = 0.05, so 5% failed both.
Q: Using the same setup, a student has failed the written component. What is the probability this student also failed the oral component?
A: First find P(W' ∩ O') = 0.05 (from above). P(W') = 1 - 0.75 = 0.25. Then P(O'|W') = P(W' ∩ O') / P(W') = 0.05 / 0.25 = 0.200. There is a 20% chance the student also failed oral.
Q: If P(A) = 0.6, P(B) = 0.3, and A and B are independent, find P(A ∪ B).
A: P(A ∩ B) = P(A) · P(B) = 0.18 (by independence). P(A ∪ B) = 0.6 + 0.3 - 0.18 = 0.72.
These probability rules feed directly into Bayes' theorem, which reverses conditional probabilities and appears in later STAT 350 material and in every statistical inference course.
The inclusion-exclusion logic extends to more than two events. For three events A, B, C, you add single probabilities, subtract pairwise intersections, then add the triple intersection.
Conditional probability is the foundation of likelihood functions used in estimation and hypothesis testing later in the course.
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