How do I work out an unknown probability? (Sorry if I'm not specific enough details in main body)

Question

There are two blue beads and x red beads in a box. The probability that two random beads taken at random from the box are both red is 15/22, how do I work out x?

Answer 1

Hint:

There are $x+2$ beads total, hence the probability of selecting a red bead first is $\frac{x}{x+2}$

What will the probability of selecting another red bead after this be?

Answer 2

The probability of picking two red balls (by uniform picking) is:

$$p = \frac {x}{x+2} \times \frac {x-1}{x+1} = \frac {x^2 - x}{(x+2)(x+1)}.$$

You want this probability to equal $\frac {15}{22}$. Just make these two quantities equal and you find the following equation:

$$\begin{aligned} 15(x^2 + 3x + 2) &= 22(x^2-x) \\ 7x^2 - 67x - 30 &= 0 \end{aligned}$$

Which gives you the (positive) solution $x = 10$.