Conditional Marble Probability

P(B|A) with marble bag examples

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P(Red | Drawn from Bag A)

Question

You have two bags. Bag A contains 4 red and 2 blue marbles. Bag B contains 1 red and 5 blue marbles. A marble is drawn from Bag A. What is the probability that it is red?

Bag A

red: 4

blue: 2

Total: 6 marbles

Bag B (not selected)

red: 1

blue: 5

Total: 6 marbles

Step-by-Step Solution

Define Events

Let A = "marble is drawn from Bag A" (given)

Let R = "marble is red"

We want P(R|A)

Focus on Bag A Only

Since we know the draw is from Bag A, we only consider Bag A's contents:

Red marbles in Bag A: 4
Blue marbles in Bag A: 2
Total in Bag A: 6

Calculate Conditional Probability

Answer

Compare: What If We Drew from Bag B?

Bag A

66.67%

Bag B

16.67%

The bag you draw from dramatically changes the probability of getting red!

Preview: Bayes' Theorem

The reverse question is also interesting: "Given that we drew a red marble, which bag was it most likely from?" This is what Bayes' Theorem helps us calculate!