Red and Blue Marbles

Question: You have 50 red marbles, 50 blue marbles and 2 jars. One of the jars is chosen at random and then one marble will be chosen from that jar at random. How would you maximize the chance of drawing a red marble? What is the probability of doing so? All 100 marbles should be placed in the jars.

Answer: Seems tricky at first right? Given that the number of red and blue marbles are the same, you would tend to think that the odds are 50-50. You would try different combinations, such as 25 of each colored marble in a jar or putting all red marbles in one jar and all the blue in the other. You would still end up with a chance of 50%.

So lets think of a better way to distribute the marbles. What if you put a single red marble in one jar and the rest of the marbles in the other jar? This way, you are guaranteed at least a 50% chance of getting a red marble (since one marble picked at random, doesn’t leave any room for choice).  Now that you have 49 red marbles left in the other jar, you have a nearly even chance of picking a red marble (49 out of 99).

So let’s calculate the total probability.

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P( red marble ) = P( Jar 1 ) * P( red marble in Jar 1 ) + P( Jar 2 ) * P( red marble in Jar 2 )
P( red marble ) = 0.5 * 1 + 0.5 * 49/99
P( red marble ) = 0.7474

Thus, we end up with ~75% chance of picking a red marble.

Have a better solution? Let us know through the comments section!

8 Responses

  1. Debra S says:

    I agree, I put 1 marble in 1 jar. 49 marbles in the other. The Blue in the trash. The probability is 1.

  2. I also came up with this answer intuitively. It’s a good solution but this definitely isn’t a proof it’s the best possible solution.
    In order to prove this seems pretty complicated though.
    This is what I did:
    r1 is the number of red marbles in jar 1
    b1 is the number of blue marbles in jar 1
    we know r2 = 50-r1
    and b2 = 50-b1
    then by the regular conditional probability rules:
    P(picking red) = 1/2 * r1/(b1+r1) + 1/2 * ((50-r1)/(100-b1-r1))

    if we let x, y = r1, b1 respectively and we know they lie in a range of [0…50].
    then z = P(picking red)

    The equation for the probability is now:
    def f(x,y):
    return 1/2 * x/(y+x) + 1/2 * ((50-x)/(100-y-x));

    plotting that in 3d shows you that there are indeed 2 maxima close to 0.75.
    I used sage to do this, and used the ranges 0…49 and 1…50 since there are discontinuities when you divide by zero.
    P = plot3d(f,(0,49),(1,50), adaptive=True, color=rainbow(60, ‘rgbtuple’));
    here’s a pic:

    Now to we need to use calculus to prove that the maxima are inded global maxima and not local maxima. We can find them using the method of gradient descent/ascent. Where the path of greatest ascent/descent on the curve has a tangent vector at any point that coincides with the gradient vector at that point.

    GRAD_V = d/dx([f(x,y)])*i + d/dy(f[x,y])*j
    (i and j are the unit vectors along x and y.)

    ie. its partial derivatives with respect to x and y must coincide with the above.

    So thats as far as I got, I know that the method of gradient ascent/descent isn’t guaranteed to converge to global maxima/minima it might get stuck and local maxima/minima. But I think we can get around this by finding roots and using the second derivative test / etc.
    Not 100% sure about that part though.

    There must an easier way to prove this else why would they ask it? I mean otherwise you’re just submitting a clever hunch as your solution.
    Let me know what you think.
    Or if you can prove that they are global maxima/minima without waving your hands in the air.


  3. Nitin says:

    Damn! I came across this questions, and took a very long time to answer (with some hints from teh interviewer!)

    wish I discovered this blog on the day it started! :)

  4. P K says:

    it’ a bit of trick question. it depends on how many times a marble is getting drawn.

  5. Brian B. says:

    I could physically feel mind mind wrench when I read this beautiful solution. Awesome!

  6. Doggie says:

    Hey John, smart answer but it clears says in the problem that “All 100 marbles should be placed in the jars”. I guess you have to think again now :)

  7. jason says:

    I believe this is the best result we can achieve

  8. John S. says:

    I would put one red marble in jar 1, 49 red marbles in jar 2, and the blue marbles in the trash. The probability of picking a red marble is now 1.

    I don’t see anything in the problem statement that prevents this, so I win.

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