Given 50 holes, what's the chance 2 balls fall into same hole In a game, a ball can fall any of $50$ holes evenly spaced around a wheel. The chance that a ball falls into any particular hole is $\dfrac 1{50}.$  What is the chance $2$ balls circling the wheel at the same time fall into the same hole?
I thought the answer would be $\left(\dfrac 1{50}\right)^2$, but the answer in the book says its still $\dfrac{1}{50}$. Can someone please explain?
 A: There are fifty holes, and for one particular hole, the probability of two balls both falling into it is $\dfrac 1{50^2}$. 
But since there are fifty holes, each with a probability of $\dfrac 1{50^2}$ of having two balls drop into it, we have an overall probability of $$\overbrace{\underbrace{\left(\dfrac 1{50}\right)^2 + \left(\dfrac 1{50}\right)^2 + \cdots + \left(\dfrac 1{50}\right)^2}_{50\;\text{ holes}}}^{1\text{st hole}\qquad 2\text{nd hole}\qquad\cdots\qquad 50\text{th hole}} =\quad 50\cdot \frac{1}{(50)^2} = \dfrac 1{50}$$
that both balls will fall in the same hole.
A: The probability is $\frac{1}{50}$ that a ball will fall into a particular hole. In this problem, however, no hole has been selected until the first ball falls into a hole. Thus, once the first ball has fallen into a hole, which happens with probability $1$, there is one hole with that ball in it, and $49$ without, so the probability that the next ball falls into the same hole is $\frac{1}{49 + 1} = \frac{1}{50}$.
That is, the problem is not what the probability is that the two balls both fall into some particular hole, but the probability that, once the first ball has fallen into a hole, the second one falls into the same hole.
A: It doesn’t matter into which hole the first ball falls: there is still just one chance in $50$ that the second ball will fall into that same hole.
If you prefer, you can work out all of the possibilities. There are $50^2$ possible ordered pairs of holes, one for the first ball and one for the second. In $50$ of those $50^2$ cases the two balls fall into the same hole, so the probabability that they fall into the same hole is $$\frac{50}{50^2}=\frac1{50}\;.$$
