Algebraic probability There are only purple and orange marbles in a bag. There are three more purple marbles than orange marbles in the bag. Roxanne is going to take at random two marbles from the bag. The probability that Roxanne will take two marbles of the same colour is 41/81. Work out the number of orange marbles in the bag.
This is my working and I don’t seem to get a proper solution. However there might be a mistake with the question as it almost works, but not quite. Can anyone explain where I went wrong?


As you suspected, the marbles are being replaced. This means the probability for the second marble is the same as the first marble.
Letting $x$ be the number of purple marbles:
$$\left( \frac{x}{2x+3} \right)^2 + \left( \frac{x+3}{2x+3} \right)^2 = \frac{41}{81}$$
$$\frac{2x^2+6x+9}{4x^2+12x+9} = \frac{41}{81}$$
$$162x^2 + 486x + 729 = 164x^2+492x+369$$
$$2x^2 + 6x -360 = 0$$
and you can continue.
