## Equal Chance

#### Problem

A bag contains `n` discs, made up of red and blue colours. Two discs are removed from the bag.

If the probability of selecting two discs of the same colour is 1/2, what can you say about the number of discs in the bag?

#### Solution

Let there be `r` red discs, so P(RB) = `r`/`n` (`n``r`)/(`n`1), similarly,

P(BR) = (`n``r`)/`n` `r`/(`n`1).

Therefore, P(different) = 2`r`(`n``r`)/(`n`(`n`1)) = 1/2.

Giving the quadratic, 4`r`^{2} 4`nr` + `n`^{2} `n` = 0.

Solving, `r` = (`n``n`)/2.

If `n` is an odd square, `n` will be odd, and similarly, when `n` is an even square, `n` will be even. Hence their sum/difference will be even, and divisible by 2.

In other words, `n` being a perfect square is both a sufficient and necessary condition for `r` to be integer and the probability of the discs being the same colour to be 1/2.

Prove that `n`(`n`+1)/2 (a triangle number), must be square, for the probability of the discs being the same colour to be 3/4, and find the smallest `n` for which this is true.

What does this tell us about `n` and `n`(`n`+1)/2 both being square?

Can you prove this result directly?