Mental Gymnastics - Answers

Mental Gymnastics - Answers

Marble Madness

Q The Acme Marble Company manufactures Purple marbles and Blue marbles.

A bag of their marbles may contain any combination of Purple and Blue marbles, including all Purple or all Blue marbles. All combinations are equally probable (a bag of zero Purple marbles is just as likely as a bag of X Purple marbles). I bought a bag of marbles and pulled one out at random - it was Purple. What is the probability that a second marble pulled at random from the same bag would also be Purple? A 2/3.

The math is fairly sophisticated; the best way to convince yourself is to solve the problem for the various cases (two of each color, three of each color, etc.). See below for an approach from a reader.

-- Reader Comments --

Let n be the total number of marbles originally in the bag. The problem is equivalent to the following: Suppose you have n+1 bags in front of you on the table, of which one contains 0 purple marbles, one contains 1 purple marble, ..., and one contains n purple marbles. You choose a bag at random and withdraw a marble without looking. It is purple. What is the probability that a second marble, randomly selected from the same bag without replacement, is also purple? The case n=2 has been extensively discussed in this newsgroup on many occasions, and it is well known that the answer is 2/3. There are 0+1+2 = 3 purple marbles in the three bags, and each of them is equally likely to be the first marble chosen (on the hypothesis that the first marble is purple). Two of the purple marbles have another purple marble in the same bag, and one has a non-purple marble. Therefore, the probability of the second marble being purple is 2/3. Now consider the general case. There are exactly n*(n+1)/2 purple marbles in the n bags, each of which is equally likely to be chosen as the first marble. For each of these, there are exactly n-1 ways to choose a second marble from the same bag, making a total of N = n*(n+1)*(n-1) / 2 equally probable ways of choosing two marbles from the same bag such that the first marble is purple. The number of ways of drawing two purple marbles from any one bag is given by P_2 = sum_k=1^n k*(k-1) = sum_k=1^n (k^2-k) = n*(n+1)*(2*n+1)/6 - n*(n+1)/3 = n*(n+1)*(2*n+1-3) / 6 = n*(n+1)*(n-1) / 3 = 2/3 * N and therefore the probability that the second marble is purple is 2/3, regardless of n.

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