On a table there are two bags. The first contains one white marble. The second bag contains one red and one white marble. Blindfolded, Billy takes one marble out of the second bag and puts it in the first bag. He then takes off his blindfold and takes a marble out of bag one. It is a white marble.
What is the probability that the other marble in the first bag is also a white marble?

Dutch National Science Quiz 12/24/2004

I saw a similar problem before so I know what the answer is (I think). I did not come up with it myself.

There are two possible outcomes for the marbles in bag 1. It's either both white (W,W) or 1 white and 1 red (W,R). Since we are told that one of them is white, this deals with conditional probability.

The white marble could come from three cases. It could be W1 from (W1,W2). It could also be W2 from (W1,W2). The third possibility is W from (W,R). Of these three cases, two of them have the other marble being white. Hence the answer is 2/3.

 

On the side, one can also say that the answer is 1/2 because the remaining marble is either red or white. Given that assumption, the answer may be argued as "correct."

If the marbles are distinguishable, then the answer would be 2/3. If they are indistinguishable, the answer will be 1/2. The only way to see whether it's 1/2 or 2/3 is to do a frequency test (simulating the test many times). I haven't run the tests myself, but I believe that 2/3 is the solution sought after.

---Edited. Changed it from black and white to red and white.

Edited on April 17, 2005, 5:09 pm

Posted by np_rt on 2005-04-17 16:44:44