# Puzzles/Statistical puzzles/3 Bags of Marbles/Solution

2/3, or approximately 66.7%

## Reasoning

The most common wrong answer is 50%. This is due to the misconception that you are given the information that you have picked either bag #1 or bag #3 [Challenge: This is not a misconception - This information is already provided in the question - a fact] . This is not the case, as explained below:

Label each of the marbles: bag 1 contains 1a and 1b, bag 2 contains 2a and 2b, and bag 3 contains 3a (white) and 3b (black). Because you picked a random bag and a random marble, each marble has an equal chance of being picked. Given that you picked a white marble, you have the following distribution:

First marble:

• 1a: 1/3
• 1b: 1/3
• 2a: 0 (black ball was not picked)
• 2b: 0 (black ball)
• 3a: 1/3
• 3b: 0 (black ball)

Given this information, we look at each situation:

• 1a: You have picked bag #1, so the second marble must be 1b, a white marble.
• 1b: You have picked bag #1, so the second marble must be 1a, a white marble.
• 3a: You have picked bag #3, so the second marble must be 3b, a black marble.

If you are in the first two situations, your second marble will be a white marble, therefore, adding 1/3 and 1/3, you get 2/3 chance of getting a white marble

## Algebraic explanation

Let W be the event of drawing a white marble, WW be the event of drawing from bag 1 (two white).

Then P(WW | W) = P(WW ∧ W) / P(W) = P(WW) / P(W) = (1/3) / (1/2) = 2/3.
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