Problem 100

Arranged Probability

If a box contains twenty-one coloured discs, composed of fifteen blue discs and six red discs, and two discs were taken at random, it can be seen that the probability of taking two blue discs, $P (B B) = (15 / 21) \times (14 / 20) = 1 / 2$ .

The next such arrangement, for which there is exactly $50 %$ chance of taking two blue discs at random, is a box containing eighty-five blue discs and thirty-five red discs.

By finding the first arrangement to contain over $10^{12} = 1, 000, 000, 000, 000$ discs in total, determine the number of blue discs that the box would contain.

设计概率

若盒子中装有 $21$ 个彩色碟子，其中 $15$ 个是蓝的， $6$ 个是红的，则随机从盒子中取出两个碟子恰好都是蓝色的概率是 $P (B B) = (15 / 21) \times (14 / 20) = 1 / 2$ 。

下一组恰好有 $50 %$ 概率取出两个蓝色碟子的情况，发生在盒子中装有 $85$ 个蓝色碟子和 $35$ 个红色碟子时。

若盒子中装有超过 $10^{12} = 1, 000, 000, 000, 000$ 个碟子，找出满足上述要求且盘子数最少的情况，并求此时盒子中蓝色碟子的数量。