Events as Subsets of a Sample Space

Lesson 1 of 2: The Language of Probability

In this lesson:

• See an event as a set of outcomes
• Combine events with "or" and "and"

What You Will Learn in This Pair

By the end of this two-lesson pair, you should be able to:

1. Identify the sample space and an event as a subset
2. Describe an event by category and by list
3. Form the union, intersection, and complement of events
4. Represent events with Venn diagrams and rosters
5. Translate everyday descriptions into set expressions

An Even Roll Names How Many?

When you say "I rolled an even number," how many outcomes are you talking about?

• One outcome? Or several?
• On a die, "even" covers 2, 4, and 6

An event can name several outcomes at once. Hold that thought.

Start by Listing Every Outcome

Roll one fair die. What can happen?

• This complete list is the sample space, written
• It holds every possible outcome — nothing left out

The sample space is your starting universe for any probability question.

An Event Is a Subset of S

An event is any subset of the sample space — a collection of outcomes.

The same event, two ways: "an even number" (characteristic) = (roster).

Name It by Category or by List

Translate the same event both ways:

Each answer is a list of outcomes — never a number.

The Idea Scales to a Card Deck

Draw one card from a 52-card deck. Now has 52 outcomes.

• "A heart" is a subset of 13 cards
• "A face card" is a subset of 12 cards (J, Q, K)

An event is still a subset — even when is large.

Quick Check: List the Event

On a die, list the event "a number less than 3."

Write it as a roster — a set of outcomes — before the next slide.

Answer:

Two Events Raise Two Questions

Define two events on the die:

• = "even" =
• = "greater than 3" =

Notice 4 and 6 are in both. Which outcomes are in either? In both?

Picture Two Events on a Venn

The sample space is the rectangle; each event is a circle.

The circles overlap because 4 and 6 belong to both events.

Union: "Or" Collects Either Event

The union is every outcome in , in , or in both.

• Shared 4 and 6 are listed once
• "Or" is inclusive — both-event outcomes count

Inclusive "or" is the idea everyone trips on.

Shade the Union: Both Circles

"A or B" shades both circles fully — a large set.

Predict First: Is 4 Included?

We're forming "even or greater than 3." The outcome 4 is in both events.

• A. Yes — 4 is part of "even or greater than 3"
• B. No — 4 is in both, so it shouldn't count

Commit to A or B before advancing.

Intersection: "And" Keeps Shared Outcomes

The intersection keeps only outcomes in both events.

• Only 4 and 6 are in both events
• Smaller than either — "and" narrows down

Inclusive "or" kept 4; "and" keeps only the overlap.

Shade the Intersection: Overlap Only

"A and B" shades only the overlap — a small set.

Cards: Heart and Face Card

Let = "a heart," = "a face card."

• and = the 3 heart face cards (J, Q, K of hearts)
• or = all 13 hearts + 9 other face cards = 22 cards

When Events Cannot Both Happen

Some event pairs have no overlap — they can't both happen.

• "Roll a 2" and "roll a 5": no outcome is in both
• These are mutually exclusive — the intersection is empty

With no overlap, inclusive and exclusive "or" agree. The overlap is what made "or" tricky.

Your Turn: Union and Intersection

On a die: = "an odd number," = "less than 4."

• List or (the union)
• List and (the intersection)

Write both rosters yourself before checking. Answers: or ; and .

Watch Out: Two Common Traps

Exclusive-or trap: "or" drops the overlap. Inclusive "or" keeps it.

Combine trap: "and" means "put it all together." No — "and" is both at once, the smaller set.

Predict which set is bigger before you compute.

Key Takeaways and What's Next

✓ An event is a set of outcomes — never a number
✓ Or = union = bigger set (overlap included)
✓ And = intersection = smaller set (overlap only)

Inclusive "or" keeps the overlap; "and" narrows it

Next: the complement "not," then combining all three operations.