Intersection: Marbles

P(A ∩ B) with marble examples

Calculator Dice Coins Marbles Cards Learn

Two Red Marbles2 Red Red Then BlueR then B Specific SequenceSequence

Drawing Two Red Marbles

What does "both red" mean?

We need two events to BOTH happen:

• First marble drawn is red
• Second marble drawn is red

In probability notation: P(1st red AND 2nd red) = P(R₁ ∩ R₂)

What does "without replacement" mean?

After drawing the first marble, we keep it out of the bag. The bag now has one fewer marble! This changes the probabilities for the second draw.

Contrast with replacement: If we put the marble back, the bag would reset to its original state and the draws would be independent.

Why does this matter?

Without replacement, the first draw affects the second draw. If we draw a red marble first, there are fewer red marbles left for the second draw. This is called conditional probability.

red: 4

blue: 3

Total: 7 marbles

Step-by-Step Solution

Calculate P(first marble is red)

Initially: 4 red + 3 blue = 7 marbles. Red marbles = 4.

Calculate P(second red | first was red)

After removing one red marble: 3 red + 3 blue = 6 marbles remain. The "|" means "given that" — we assume the first was red.

Multiply (for dependent events)

For "A AND B" with dependent events: P(A ∩ B) = P(A) × P(B | A)

Result:

Key Insight

Without replacement makes each draw dependent on previous draws. The formula P(A ∩ B) = P(A) × P(B | A) accounts for this dependency, where P(B | A) is recalculated based on what's left after A happens.