Number of Marbles to Remove Allowing 2 Marbles of Each Color

Discussion in 'Homework Help' started by zulfi100, Aug 19, 2017.

  1. zulfi100

    Thread Starter Active Member

    Jun 7, 2012
    458
    1
    Hi,
    I have got following problem unsolved:

    A jar contains 20 marbles : 4 red , 6 white and 10 blue. If you remove marbles one at a time, randomly, what is the minimum number that must be removed to be certain that you have at least 2 marbles of each color?

    I think 14 marbles must be removed. This would leave us with only 6 marbles remaining and then we can be certain that we have atleast 2 marbles of each color.
    However my answer is not correct. Somebody please guide me.

    Zulfi.
     
  2. MrAl

    AAC Fanatic!

    Jun 17, 2014
    5,189
    1,120
    Hi,

    Well, if you remove 4 red and 6 white by pure chance, how many of what color would be left and how many more would you have to pick out? I think that establishes a lower limit with some bad luck.
    But, if you picked out 6 white and 10 blue, you still would not have any red so how many more would you have to pick? That kind of says that 14 isnt right.

    Try to imagine the overall process as you pick out marbles one by one and what the possibilities are. You have to consider that you may have very bad luck as you want to know the maximum picks required.
     
  3. WBahn

    Moderator

    Mar 31, 2012
    22,990
    6,884
    Your statement of the problem is a bit vague. When you say, "you have at least two marbles of each color," does that mean that you've withdrawn two marbles of each color, or that there are still at least two marbles of each color left in the jar?

    I'm guessing (and please correct me if I'm wrong) that you mean that you are certain to have removed at least two marbles of each color.

    So let's consider your answer of 14 marbles, leaving six in the jar. What if the six marbles in the jar are all white? Then you have removed four red marbles and ten blue marbles. Have you removed at least two marbles of each color?

    This problem really isn't about probabilities (though it can be cast in that form). You are not asked for a probability, you are asked for a certainty. A guarantee. In terms of probabilities, you are asked for either a probability of 0% or a probability of 100%, depending on how you phrase the question.

    So ask yourself what is the maximum number of marbles you can place in your hand without having at least two of every color. The number you are looking for is then one more than that.
     
  4. Raymond Genovese

    Active Member

    Mar 5, 2016
    973
    537
    No. If you remove 14 marbles consisting of 10 blue and 4 whites, you do not have two of each color. Because the question states "to be certain you have at least 2 marbles of each color", you have to consider the worst possible picks with regard to getting 2 of each color. The worst possible picks in this case is picking all of the blues and whites first.
     
  5. zulfi100

    Thread Starter Active Member

    Jun 7, 2012
    458
    1
    Hi,
    Thanks all. This is a matter of English. Question says:
    Does it mean only 2 marbles of one color?
    or
    2 marbles of all color?

    If it means only 2 marbles of one color then we have to remove 18 marbles. It does not ask for the color. So still we dont know that both the marbles removed are of same color but there is a possibility that they can be of same color.
    Am i right?

    Zulfi.
     
  6. Raymond Genovese

    Active Member

    Mar 5, 2016
    973
    537
    You stated the original question as:
    "A jar contains 20 marbles : 4 red , 6 white and 10 blue. If you remove marbles one at a time, randomly, what is the minimum number that must be removed to be certain that you have at least 2 marbles of each color?"

    That means at least two red marbles and at least two white marbles and at least two blue marbles.

    Your answer of 18 is correct. In ALL cases, if you randomly draw 18 marbles, you will have at least two of each color.

    You COULD get two of each color in as few as 6 picks, but to solve the problem, you have to consider the worst case possibility and still be CERTAIN that you have at least two of each color.

    Picking all 6 white and all 10 blue on your first 16 picks means that the 17th and 18th picks will both be red to satisfy the request. Correct answer=18.

    Picking all 6 white and all 10 blue and 1 red on your first 17 picks (regardless of where in the 17 picks the red appears), means that your 18th pick will be the second red to satisfy the request. Correct answer 18.

    "So still we dont know that both the marbles removed are of same color but there is a possibility that they can be of same color."

    That is another question, but IF you pick all 6 white and all 10 blue on your first 16 picks, you do know that the next two picks will be red. In any case other than picking all of the 6 white and 10 blue on your first 16 picks, you do not know the color of the next two marbles.

    But, again, the question is not about the possibility of getting at least two of each color, it is about the certainty of getting at least two of each color.
     
    Last edited: Aug 27, 2017
    zulfi100 likes this.
  7. WBahn

    Moderator

    Mar 31, 2012
    22,990
    6,884
    The statement, "at least 2 marbles of each color," is unambiguous. It means two or more red marbles AND two or more white marbles AND two or more blue marbles.

    The phrase, "only two marbles of one" would mean, "exactly two marbles of one color".

    That's not something that we can be certain of because we COULD achieve that after drawing two marbles and then violate it upon drawing a third. Or, we might have to draw as many as four marbles before we first had two marbles of one color.
     
  8. zulfi100

    Thread Starter Active Member

    Jun 7, 2012
    458
    1
    Hi,
    Thanks for removing my confusion. But still i got confused
    I though it was asking for the remaining 2 marbles left in the jar. Any way now i am able to understand that it is related to marbles outside of the jar which are removed.
    but in the above case there is a possibility that all the six marbles are white.
    Another chance is that we have 16 marbles. In this case there is a possibility that we remove 6 white and 10 blues but this is not fine because we dont have 2 reds.
    Another chance is that we have 17 marbles outside. In this case there is a possibility that we remove 6 white and 10 blues and 1 red but this is not fine because we dont have 2 reds.
    Another chance is that we have 18 marbles outside. In this case there is a possibility that we remove 6 white and 10 blues and 2 red and this is fine because we have 2 reds, atleast 2 white and atleast 2 blues. This is required in the question.

    God bless you people for giving me so much time.
    Zulfi.
     
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