Intersection: Dice

P(A ∩ B) - outcomes satisfying BOTH conditions

Calculator Dice Coins Marbles Cards Learn

Even AND Greater Than 3Even & GT3 Prime AND OddPrime & Odd Less Than 4 AND OddLT4 & Odd

What Does "AND" Mean in Probability?

Plain English Definition:

"A AND B" means: both conditions must be true at the same time. The outcome must satisfy condition A and also satisfy condition B.

Think of it as a filter:

Start with all outcomes, then filter to keep only those that pass both tests. Each additional "AND" condition makes the set smaller (or stays the same).

Why is Intersection Often Smaller?

Intuition:

The more requirements you add, the harder it is to satisfy all of them. "Being tall" has many people. "Being tall AND being a doctor" has fewer. "Being tall AND being a doctor AND being left-handed" has even fewer.

Mathematical fact:

and

The intersection can never be bigger than either individual event.

Example: Rolling Even AND Greater Than 3

Question

What is the probability of rolling a number that is both even AND greater than 3?

A: Even Numbers

✓

Candidates: 2, 4, 6

B: Greater Than 3

✓

Candidates: 4, 5, 6

Finding the Intersection: Which numbers pass BOTH tests?

Go through each number and check both conditions:

• 2: Even ✓, Greater than 3 ✗ — fails second test
• 4: Even ✓, Greater than 3 ✓ — passes both!
• 5: Even ✗ — fails first test (5 is odd)
• 6: Even ✓, Greater than 3 ✓ — passes both!

A ∩ B: Numbers that are BOTH even AND greater than 3

✓

Only 4 and 6 satisfy both conditions!

Probability Calculation

Count outcomes that pass both tests

We found 2 numbers (4 and 6) that satisfy both conditions.

Divide by total outcomes

As always, probability = (favorable outcomes) / (total outcomes)

Final Answer:

There's a 1 in 3 chance of rolling a number that is both even and greater than 3.