Calculating Nth place from 1st place probability

My apologies in advance, as I'm not sure how advanced this is, this might belong in basic Probability and Statistics.

I've done a lot of searching online, and I can't seem to find anything to say this problem has ever been attempted mathematically (which is the reason I'm guessing this would count as advanced).

The problem is this, lets say you have a race with 6 people in it. Given you know the chance of them to come first, can you mathematically calculate their chance to come 2nd (and 3rd, 4th, 5th, last)?

So for example:

Runner 1: 25% chance to win

Runner 2: 20% chance to win

Runner 3: 20% chance to win

Runner 4: 15% chance to win

Runner 5: 12% chance to win

Runner 6: 8% chance to win

If you are able to build a formula, would you be able to build it with a variable number of runners?

Thanks in advance to those of you who try. :-)