2/3, or approximately 66.7%
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:
- 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
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.
Correct answer - 50% - is commonly considered wrong, but really - past events have no influence on future probabilities at all (You cannot count "You have picked bag #1" twice. You just either did it or not). Right now You have one white marble in your hand and knowledge, that another one is either black or white depending on which bag (out of two) you selected.
Challenge: You essentially just start by picking a marble--and there are three white marbles. Two of those white marbles are in the same bag as another white marble; so, given that you've picked one of the three white marbles, 2/3 of the time its pair matches--right?