It's possible that my friend will say it's pretty unlikely, after all she's just taken 99 balls out at random and they were

**white. She's sort of right, the likelihood is remote at 1%, but it's possible that she might think it's even more improbable. She'd be wrong though, wouldn't she?**

*all*One way we could think of it is "what is the likelihood of my friend picking the only non-white ball in the 100th position, rather than one of the other positions?" Well, there are 100 positions in a random sequence of 100 balls in which a lone non-white ball could lie, so the likelihood of it lying in the 100th position is the same as it lying in the first, or any other position namely 1/100. Therefore, either the last ball still in the urn is non-white (at a probability of 1/100), or it's white just like all the others.

This logic should work the same no matter how I fill the urn and no matter how many balls I put in the urn. If there are a million balls, the chances of the last ball being the only non-white one after pulling out 999,999 white balls is one in a million. If there are two balls, then the chances of the second ball being non-white after the first being white is 50%.

But wait a minute, let's look at my process for filling an urn with two balls.

Let's say that my selection process involves a million different balls from which I make my selection, only of which is white, the rest having different colour and pattern combinations. I select one ball completely at random and put it in the urn. The ball I removed is replaced by an impartial third party with an identical one, and then I make another entirely random selection and put it in the urn.

What happens to the probability then? Say that my friend pulls out a ball, perfectly at random, and it is white. What are the chances that the one left in the urn will be white?

I encourage people to think carefully about this little conundrum, provide an answer and give a brief explanation.