A bowl contains only red and green marbles. The probability of selecting two marbles of the same color from this bowl is equal to 1 half. Which of the following is a possible number of marbles in this bowl? Let's call the number of marbles in the bowl M. And that is made up of adding the red and the green marbles, which I will call R plus G. And so then the probability of choosing a red marble would be R over M. And the probability of choosing a second red marble would be the product here of R minus 1 divided by M minus 1. We have two events and we multiply them together like this. And likewise I can do the same thing for the probability of choosing a green marble two times in a row and together that would be equal to 1 half. Now we will change the variable here to introduce the variable M. So I have a relationship between R, G, and M that I will use and I will say that R is equal to M minus G. And then substitute that into the equation. So I have M minus G over M times M minus G minus 1 over M minus 1 plus and then the old equation here G over M G minus 1 over M minus 1 is equal to 1 half. And then I will keep working with this first giving a common denominator to this. So multiply both sides by 2 times M times M minus 1. And so when we do that the M times M minus 1 cancel in the denominator but I have an extra factor of 2 on the left hand side and on the right hand side the 2 cancels and so I have an extra product here of M and M minus 1. And then I will simplify this. So that's 2 and then let's multiply everything together. The first product will give me M squared minus M G minus M minus G M plus G squared plus G. And the second product is G squared minus G and that's all multiplied by 2. On the right hand side I have a M squared minus M. So then simplifying more inside the parentheses here there is a M squared 2 M times G negative 2 G squares the powers single powers of G cancel and then I have a minus M left over that's equal to M squared minus M. So then finally bringing everything to the left hand side I have a 4 G squared 1 power of M squared cancels plus M squared minus 4 M G and then the power of M will cancel and I will put that on the right hand side. So we have this following expression for the number of marbles in the bowl M and after staring at this for a while we note that M here is a perfect square. I can rewrite the left hand side slightly and we have that this is 2 G plus M quantity squared which we can quickly check and so this tells us that M here is a perfect square. So the number of marbles in the bowl have to be chosen so that it's perfect square looking at our choices here the only choice that is a perfect square is the first one 81 so that has to be the number of marbles.

marbles

probability

perfect square

algebraic manipulation

probability calculation