Bayes' Theorem with Marbles

Update beliefs about bags based on draws

Calculator Dice Coins Marbles Cards Learn

Which Bag?Which?Bag IdentificationIdentify Sequential DrawsSequence

Updating Belief After Multiple Draws

The Scenario

You have two bags. You pick one at random, then draw marbles one at a time (with replacement). After each draw, update your belief about which bag you have.

The Two Bags

red: 7

blue: 3

Total: 10 marbles

Bag A (red-heavy)

P(red) = 70%, P(blue) = 30%

red: 3

blue: 7

Total: 10 marbles

Bag B (blue-heavy)

P(red) = 30%, P(blue) = 70%

Prior:

Sequence: Red, Blue, Red, Red

Draw 1: Red

Starting with P(A) = 0.5:

P(Bag A)

70%

P(Bag B)

30%

Draw 2: Blue

Now P(A) = 0.70:

P(Bag A)

50%

P(Bag B)

50%

Draw 3: Red

Back to P(A) = 0.50:

P(Bag A)

70%

P(Bag B)

30%

Draw 4: Red

With P(A) = 0.70:

P(Bag A)

84.5%

P(Bag B)

15.5%

Belief Evolution

Draw	Color	P(Bag A)	P(Bag B)
Before	-	50.0%	50.0%
1	Red	70.0%	30.0%
2	Blue	50.0%	50.0%
3	Red	70.0%	30.0%
4	Red	84.5%	15.5%

Key Insights

1.Each draw updates our belief based on the previous posterior
2.With 3 red and 1 blue (75% red), we strongly favor Bag A (84.5%)
3.Blue draws push toward Bag B, red draws push toward Bag A
4.More draws give more confidence in the conclusion