Conditional Marble Probability

P(B|A) with marble bag examples

Calculator Dice Coins Marbles Cards Learn

P(2nd Red | 1st was Red)

Understanding P(B | A) for sequential draws:

P(2nd Red | 1st Red) asks: "What's the probability the second marble is red, if we already know the first marble was red?"

Why does the first draw matter?

Because we're drawing without replacement! When we remove the first red marble, the bag composition changes — there's now one fewer red marble available. This is what makes the events dependent.

Question

A bag contains 5 red and 3 blue marbles. You draw one marble, keep it out, then draw another. Given that the first marble was red, what is the probability that the second is also red?

red: 5

blue: 3

Total: 8 marbles

Step-by-Step Solution

Initial Setup

Red marbles: 5

Blue marbles: 3

Total: 8 marbles

After First Draw (Given: It Was Red)

Since we drew a red marble and kept it out:

Red remaining: 4

Blue remaining: 3

Total remaining: 7 marbles

Identify Favorable Outcomes for Second Draw

We want the second marble to be red.

Red marbles available: 4
Total marbles available: 7

Calculate Conditional Probability

Answer

Key Insight: Dependent Events

Notice how the first draw affects the second:

Unconditional P(2nd Red)

Would need to consider all first draw outcomes

Conditional P(2nd Red | 1st Red)

Drawing without replacement creates dependent events - each draw changes the composition of the bag.