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| Math Central | Quandaries & Queries |
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A jar has 16 marbles in it. Most of the marbles are Red and the rest are White. Two marbles are taken out of the jar at the same time. It is equally probable that two marbles are of the same colour as the two marbles of different colour (that is the probability that they are both red or both white is the same as they are different). How many marbles are Red and how many marbles are White? Could anyone help me with the above question? Best regards, |
Hi Angie,
This is a nice problem. First, we can notice that drawing two marbles at the same time is the same as drawing first one marble, then another, so long as the first marble is not put back in the jar.
Let r be the number of red marbles. In that case, there are 16-r white marbles. There are a couple of cases to consider. First we consider if both marbles drawn are red. The probability that the first marble is red is r/16 (number of red marble/total number of marbles). The probability that the second is red, given that one red marble has been removed, is (r-1)/15. Thus the probability of drawing two red marbles is r(r-1)/(15 
16)).
Secondly use a similar procedure to find the probability that both marbles drawn are white. Then the probability that both marbles are the same colour is the probability that they are both red plus the probability that they are both white.
But the only other possibility is that the marbles are not the same colour. Since we know these probabilities are the same, and must sum to 1, we know that they must both have probability of 1/2. Hence the probability that they are both red plus the probability that they are both white is 1/2.
Solve the resulting expression for r, the number of red marbles.
Danny
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