Bayes' Theorem with Marbles

Update beliefs about bags based on draws

Calculator Dice Coins Marbles Cards Learn

Which Bag?Which?Bag IdentificationIdentify Sequential DrawsSequence

The Classic Two-Bag Problem

red: 7

blue: 3

Total: 10 marbles

Bag A

7 red, 3 blue

red: 3

blue: 7

Total: 10 marbles

Bag B

3 red, 7 blue

The Scenario

You have two bags. Bag A has 7 red and 3 blue marbles. Bag B has 3 red and 7 blue. You pick a bag at random, draw one marble, and it's red. What is the probability you picked Bag A?

Problem Setup

Bag A (red-heavy)

(picked at random)

Bag B (blue-heavy)

(picked at random)

Step-by-Step Solution

Calculate P(red) - Total Probability

Apply Bayes' Theorem

Answer

Given you drew a red marble, there is a 70% chance you picked Bag A.

P(Bag A | red)

70%

P(Bag B | red)

30%

Building Intuition

Imagine doing this experiment 100 times:

Pick Bag A (50 times), draw red:50 × 0.7 = 35 times

Pick Bag B (50 times), draw red:50 × 0.3 = 15 times

P(Bag A | red) =35/50 = 70%