When I wrote the last post on the "Two Balls, One Urn" scenario, Lokee had a look, pointed out some typos and said "It's too confusing!" I tried to explain a little more simply, but it might still be a little unclear.
Fortunately, as the "creator" I am not obliged to maintain precisely the same scenario to get at the point so, let me suggest a slightly different scenario:
I have an enormous barrel in a pitch black room and I know that in the barrel there are two million balls, only two of which are entirely white, the rest being of various colours, shades and and patterns. I take my urn into the pitch black room and, completely at random, I take out two balls from the barrel and place them in the urn. Because it is so dark in the pitch black room, I cannot see either of the balls when I do this.
Then, while still in the pitch black room, I reach into the urn and, completely at random, I draw out one ball. Because it is so dark in the pitch black room, I cannot see either of the balls when I do this.
Finally I walk out of the pitch black room and I look at the ball I drew out of the urn. It is white.
What is the probability that the second ball - the one still in the urn - is white?
Suppose I filled an urn with 100 balls and, having not let my friend see this process, ask her to take out the balls one by one - at random. After 99 balls, I ask her to stop. Now, suppose further that she had taken out 99 white balls. I ask her what the likelihood is that the ball still in the urn is non-white.
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 all 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?
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 one 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.
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