The Addition Rule: Disjoint Events and Tables

Lesson 2 of 2: One Rule, Every Case

In this lesson:

• Apply the rule when events can't both happen
• Compute and interpret "or" queries from tables

What You Will Be Able to Do

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

1. Apply the special case for disjoint events
2. Decide whether to subtract by asking "can both happen?"
3. Compute and interpret an "or" probability from a two-way table

P(2 or 5): Do We Subtract Here?

Roll a fair die. Find .

Last lesson we subtracted an overlap. Do we subtract one here?

Think about whether a single roll can be both a 2 and a 5.

Ask First: Can These Both Happen?

Before applying the rule, ask one question:

Can both events happen at once?

• No → no overlap → nothing to subtract
• Yes → overlap exists → subtract it once

Mutually Exclusive Means No Overlap

• A single roll can't be both a 2 and a 5
• These events are mutually exclusive (disjoint)
• Their Venn circles don't touch

No Overlap Means the Overlap Is Zero

If the circles don't touch, there are no shared outcomes:

There is literally nothing in the overlap to subtract.

Apply It: P(2 or 5) = 1/3

The full rule, with a zero overlap:

Same rule as last lesson — the subtracted term is just zero.

Same Rule — The Term Is Just Zero

There is one Addition Rule:

• Overlapping events: subtract a real overlap
• Disjoint events: that term is 0, so you just add

Predict: Is "Heart or Red" Disjoint?

Draw one card. Are "draw a heart" and "draw a red card" mutually exclusive?

• A. Yes — they're separate events
• B. No — they can both happen

Commit to A or B before advancing. This one fools people.

Classify: Which Pairs Can Both Happen?

For a single card draw, decide if each pair is disjoint:

• Draw a king or a queen
• Draw a face card or a heart

Decide each before advancing — then you'll know whether to subtract.

Now Read the Pieces Off a Table

Real "or" questions usually arrive as data in a two-way table.

A table hands you , , and directly — from its margins and cells.

Next: one table, one "or" query, computed two ways.

Read an "Or" Query From a Table

Find by counting the union students directly.

Same Query by the Addition Rule

From the table's margins and the joint cell:

Both Routes Agree on the Answer

Direct union count and the Addition Rule both give:

• The joint cell is the overlap
• Counting union cells already avoids it; the rule subtracts it

Interpret What the Probability Means

A probability isn't finished until you say what it means:

60% of these students either buy lunch or are seniors (or both).

Always pair the number with a sentence in the model's context.

Your Turn: A Table Query, Two Ways

Using the same table, find .

Compute it two ways, then write a one-sentence interpretation.

Direct count and the rule should agree — then say what it means.

Two Common Errors to Watch For

Subtracting when disjoint: removing an overlap that's actually zero
Ask "can both happen?" — if no, just add; there's nothing to subtract.

"Or" means multiply: swapping in the Multiplication Rule
"Or" adds (then subtracts the overlap); "and" multiplies.

Ask "Can Both Happen?" Then Interpret

✓ One rule:
✓ Disjoint → the overlap is 0, so you just add
✓ Tables give every term — always interpret the result

Next: the Multiplication Rule — the "and" companion to this "or" rule.