AP Statistics · Unit 2 · Topic 2.5

Mutually Exclusive Events Study Lab

Two events are mutually exclusive when they cannot happen at the same time, so the probability that both happen, P(A and B), is 0. Choose two events from a standard deck, see which cards are in A, in B, and in both, and practice spotting mutually exclusive events in context.

1. Can both happen at once?

Draw one card from a standard 52-card deck. Pick two events and look for cards that are in both.

P(A and B) = outcomes in both ÷ total Mutually exclusive ⇔ P(A and B) = 0
Try a pair

The deck

A onlyB onlyA and B

Venn diagram counts of cards

P(A)
P(B)
P(A and B)
Mutually exclusive?

Show your work

Practice: mutually exclusive or not?

For each pair, ask: can a single outcome be in both events?

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2. Key ideas

Joint probability

"A and B"

The event "A and B" (the intersection, A ∩ B) happens when one outcome is in both events. Its probability is the joint probability P(A and B).

Definition

Mutually exclusive

Events are mutually exclusive (disjoint) if they have no outcomes in common. Then P(A and B) = 0: they can never happen on the same trial.

Venn diagram

No overlap

In a Venn diagram, mutually exclusive events are drawn as circles that do not overlap. Overlapping circles mean some outcomes are in both.

Two-way table

Reading the cell

In a two-way table, the joint probability for a row category and a column category is that cell ÷ the grand total. A cell count of 0 means those two categories are mutually exclusive in the data.

Complements

Always disjoint

An event and its complement are always mutually exclusive, and together they cover the whole sample space. Mutually exclusive events do not have to be complements, though.

Not the same thing

Disjoint ≠ independent

Mutually exclusive events with nonzero probabilities are never independent: if one happens, the other cannot. Independence comes in Topic 2.7.

3. Worked examples

EventsAny outcome in both?P(A and B)Mutually exclusive?
Draw a card: A = heart, B = spadeNo card has two suits0Yes
Draw a card: A = heart, B = kingThe king of hearts1/52No
Roll a die: A = even, B = greater than 46 is in both1/6No
Roll a die: A = 1 or 2, B = 5 or 6No shared faces0Yes
A random student: A = junior, B = plays a sport (table shows 42 of 200 are both)42 students42/200 = 0.21No

4. On the FRQ

How to earn the points

  • Justify with a probability: "The events are not mutually exclusive because P(junior and plays a sport) = 42/200 = 0.21, which is not 0."
  • Or name a shared outcome: "A student can be both a junior and play a sport."
  • Use the context in your answer, not just "A and B overlap."

Common mistakes

  • Calling events mutually exclusive because they are "different." Heart and king are different events but share the king of hearts.
  • Mixing up "mutually exclusive" with "independent." They are different ideas, and they almost never go together.
  • Multiplying to get P(A and B) without knowing the events are independent.
  • Saying P(A and B) is 0 just because it is small.

5. Practice questions

AP-style multiple choice. Pick an answer to see the explanation.

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6. Key terms

The vocabulary for topic 2.5. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 2 terminology guide.