Simplifying a seemingly complex probability problem.

Question

We randomly put $100$ numbered balls in $100$ baskets, and if I am to ask what's the probability of the third ball being in a basket between the first and second ball, I know the answer is exactly one third, intuitively. A similar case would be asking what's the probability of one of these balls being bigger than the other. The difference is that for the latter example (with the probability being $1/2$), I can formally prove it using the law of total probability. In the first example I was not able to devise or think of a simple proof, so I was wondering if you could perhaps show me how it can be done using formality rather than common sense.

Thanks a million!

Answer 1

Assuming only one ball can go in each basket, you can ignore all the other balls and look at the order of balls $1,2,3$. There are $3!=6$ orders of these balls, and ball $3$ is in the middle in $2$ of them.