24 October 2013

Puzzle time

You are shown two identical bowls in a room – one of them containing 25 white marbles and the other containing 25 black marbles. The marbles are identical to each other other than the aspect of color, of course.
A blindfold person will be brought in to the room and asked to choose any bowl randomly and pick any one marble randomly from that bowl. If it is turns out to be a white marble, you will get a million dollars. If it turns out to be black you get nothing.
However, before the blindfolded person is brought in, you are allowed to mix up all the marbles and place any number of white and any number of black marbles in each bowl. Needless to say, all marbles have to be in one or the other bowl.
How would you place the marbles (meaning how many of which color marbles in which bowl) to give you the maximum chance of winning the million dollars?

(Remember to send me FB message instead of posting it on the Comments section. I will acknowledge you in the comments section if you get it right)

Posted October 24, 2013 by Rajib Roy in category "Puzzles


  1. By Sreenivas Makala (Post author) on

    Rajib Roy Sir,intuitively place one White Ball in one bowl and 24White and 25 Black in another bowl.I am getting the same answer if I take a generic case and try to maximise the conditional probability by first principles.Is this right?

  2. By Rajib Roy (Post author) on

    The Answer:
    To think of the problem, let’s start by the current status – 25W in one bowl and 25B in another bowl. The probability of picking any bowl is 1/2 and then the prob of picking a white for first bowl is 1 (only whites) and the other is 0 (only black). So, the combined prob is (1/2 * 1) + (1/2 * 0) = 1/2 (the events are mutually exclusive; so you add the prob).
    Now let’s try taking a black marble and put it in the bowl with only white marbles. We get (1/2 * 25/26) + (1/2 * 0). Which is less than 1/2 (original case). In fact, the more black marbles you put in the bowl with white marbles, the lesser the prob becomes. (The one that has black marbles will always be 0 and the one that has white marbles is getting lesser and lesser because we are adding black marbles).
    So, let’s go the other way. Let’s put one white marble in the bowl with black marbles. The probability is (1/2 * 1) + (1/2 * 1/26) which is higher than 1/2 (original case). As you put more and more white marbles in the bowl with black marbles, the probability goes higher and higher.
    In fact, if the bowl with white marbles is selected, it does not matter how many white marbles are there, the prob is always 1. So, you just keep 1 in that bowl and put the rest in the other bowl to maximize the chance of picking white in that bowl, should that bowl be selected.
    Therefore the answer is 1 white in one bowl and rest of the marbles in the other.
    The prob will be (1/2 * 1) + (1/2 * 24/50) = 0.74 (as the number of marbles increase, this gets closer and closer to 0.75)

  3. By Bob Hart (Post author) on

    I would have just put all but one white marble on top of all the black marbles. I would submit that, as long as you don’t stir the marbles, despite what probability says, a sampling of blind people picking marbles would return a much greater than 75% chance of winning the million dollars ๐Ÿ™‚


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