AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/2/2-5)
Unit 2 · Topic 2.5
2.5 Mutually Exclusive Events
Two events are mutually exclusive (disjoint) if they can't both happen on the same trial, so their joint probability is 0. You'll use the joint probability to justify whether events are mutually exclusive, and keep this idea separate from independence.
Key terms
- mutually exclusive (disjoint)
- joint probability
- intersection P(A ∩ B)
Joint probability
The joint probability of A and B is the probability that both happen. It's written P(A ∩ B), read "A intersect B" or "A and B."
From the grade-and-sport table in 2.1 (200 students), P(freshman ∩ plays a sport) = 72/200 = 0.36.
Mutually exclusive events
Two events are mutually exclusive (disjoint) when no single outcome belongs to both. One trial can't land in both. Then P(A ∩ B) = 0.
Examples: rolling a 2 and rolling a 5 on one die; a student being a freshman and a senior. Not mutually exclusive: being a freshman and playing a sport, since 72 students are both.
Justifying with the joint probability
To show events are mutually exclusive, show P(A ∩ B) = 0. To show they're not, show P(A ∩ B) > 0, ideally by naming an outcome that's in both.
Write it out: "Being a freshman and playing a sport are not mutually exclusive because P(freshman ∩ sport) = 72/200 = 0.36, which is not 0." Just saying "they can both happen" is weaker than giving the number.
Picturing it
A Venn diagram draws each event as a circle inside a rectangle that stands for the sample space. If the circles overlap, the overlap is A ∩ B and the events are not mutually exclusive. If the circles are drawn apart with no overlap, the events are mutually exclusive.
In a two-way table, two categories of the same variable (freshman, senior) are always mutually exclusive, because each unit is in only one row. A row category and a column category (freshman, plays a sport) are mutually exclusive only if their shared cell is 0.
Disjoint is not independent
Students often mix up mutually exclusive and independent. They're nearly opposites. If A and B are disjoint and each has positive probability, then knowing A happened tells you B definitely didn't. That's a big change in B's probability, so the events are dependent.
Keep them straight: mutually exclusive is about whether events can overlap; independent (2.7) is about whether one changes the probability of the other.
For disjoint events, the addition rule simplifies: P(A ∪ B) = P(A) + P(B), since the overlap term is 0.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Justify mutually exclusive or not
One card is drawn from a standard 52-card deck. Let A = the card is a heart, B = the card is a club, C = the card is a king. Are A and B mutually exclusive? Are A and C?
Show the solutionHide the solution
- Step 1: A ∩ B means a card that is both a heart and a club. No such card exists, so P(A ∩ B) = 0. A and B are mutually exclusive.
- Step 2: A ∩ C means the king of hearts, which exists. P(A ∩ C) = 1/52 ≈ 0.019, which is not 0. A and C are not mutually exclusive.
Answer: A and B are mutually exclusive, since P(A ∩ B) = 0. A and C are not, since P(A ∩ C) = 1/52 > 0.
- Example 2Calculator allowed
Trap: disjoint and independent?
For one roll of a fair die, let A = roll a 1 and B = roll a 6. A student says A and B are independent because they're mutually exclusive. Check the claim.
Show the solutionHide the solution
- Step 1: P(A) = 1/6, P(B) = 1/6, P(A ∩ B) = 0, so they're mutually exclusive.
- Step 2: For independence you'd need P(A ∩ B) = P(A) · P(B) = 1/36. But P(A ∩ B) = 0, not 1/36.
- Step 3: So they're dependent: if you know you rolled a 1, the chance of a 6 drops from 1/6 to 0.
Answer: The claim is wrong. A and B are mutually exclusive but not independent.
Common mistakes
- Treating "mutually exclusive" and "independent" as the same thing.
- Justifying with words only. Give the joint probability or an outcome in both events.
- Adding P(A) + P(B) for "A or B" when the events overlap. That only works for disjoint events.
On the exam
- Multiple-choice questions often ask which pair of events is mutually exclusive. Look for a pair that can't share an outcome.
- If a question asks you to justify, compute P(A ∩ B) from the table and compare it with 0.
Connected topics
Videos
Check yourself
4 questions on 2.5 Mutually Exclusive Events. Pick an answer to see if you got it, and why.
At a school, 30% of students play a fall sport and 25% are in the fall musical. Students can't do both, because practices are at the same time. What is the probability that a randomly selected student does a fall sport or is in the fall musical?
In a class of 30 students, 12 have a part-time job and 9 play in the band. Four students do both. Which statement is correct?
Events A and B are mutually exclusive, with P(A) = 0.4 and P(B) = 0.3. Which statement is true?
| Grade level | Bus | Car | Walk or bike | Total |
|---|---|---|---|---|
| Grades 9–10 | 90 | 40 | 70 | 200 |
| Grades 11–12 | 50 | 120 | 30 | 200 |
| Total | 140 | 160 | 100 | 400 |
Invented data: how 400 randomly selected students at a high school usually get to school
Which pair of events is mutually exclusive for a randomly selected student?
0 of 4 answered