Say I'm betting on multiple teams to win in a basketball tournament. Do I want my teams arranged so that they could meet as early as possible (in the first round) or as late as possible (in the finals)? I just calculated the simplest case:

A four team tournament, with seeding selected randomly. Two teams (A and B) are 2/3 favorites against the other two teams (B and C) any time they face them. So A and B each have 58/162 chance to win the tournament, and each C and D have a 23/162 chance to win the tournament. (I hope I calculated that right.) So say I bet on both C and D to win the tournament at +643, what do I hope the bracket looks like? Well, I would want C and D to be playing each other in the first round, because that would mean I have a 1/3 chance that one of them will win the tournament (each has a 1/6 chance). If they were on opposite sides of the tournament bracket, then I'd only have a 7/27 chance that one of them would win the tournament (each has a 7/54 chance).

So it seems if I'm betting on underdogs in a tournament, I want them to meet as early as possible. That must mean if I'm betting on favorites, I want them to meet as late as possible (be on opposite sides of the bracket). I would also assume that if I were betting on people to win a coin-flipping tournament, it wouldn't make any difference to me where they were seeded in the tournament bracket. Can anyone confirm that this is true for tournaments of any size and I haven't messed anything up?

I guess that you are right. Seeding was introduced exactly to give favourites an advantage. If stronger teams are distributed uniformly all across the draw, that means that an underdog has to beat several of them to win. On the other hand, if the favourites are confined in a small section of the draw, an underdog might need to win just one match over a stronger team.

Of course if every team has the same strength, the draw plays no role.

Yes, nick has it spot on.

Consider the simplest case of only 4 teams consisting of 2 evenly matched strong teams and 2 evenly matched weak teams. Suppose either strong team has an 80% chance of beating either weak team.

Then if the two strong teams are in the same initial bracket, a strong team will win the tournament 80% of the time (obviously). But if the two strong teams are in different initial brackets, then a strong team will win the tournament 89.6% of the time for the reasoning nick gave (a weak team would very likely have to beat both strong teams to win the tourney).

So if your horses are good players (favorites), you want them spread out (meet late in the tourney). And if your horses are weak players (underdogs), you want them bunched together (meet early in the tourney).