Claimed:Two players: same or different conference?

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A while back I had a conversation with @Mr. Power about which would be better to maximize your odds of winning a Continental Cup: having two players on competing teams in the same conference, or two players on competing teams in opposite conferences. For the sake of simplifying the argument, I'm going to say that each of the competing teams will have an equal shot of beating each of the other competing teams. We're also going to say we have five competing teams; two in one conference (which I shall call the 'weak conference') and three in the other (the 'strong conference').

 

I will be taking a look at three different scenarios, because if you are on two teams in the same conference, it could be either the weak conference or the strong conference. I should note that I am figuring out the odds as I'm going through this, so I don't know exactly what the results will show. I think mathematically with all five competing teams being equal, there is no strict advantage to either having your teams in the same or different conferences, with the caveat being that there is also a 50% chance of you ending up in either the weak or strong conference.

 

 

Scenario 1: Two players in weak conference

If there are two competing teams and you have a player on each of them, then you'll be guaranteed a shot in the finals. And once you are there, you have a 50% chance of winning the Cup. This is undoubtedly probably the best chance you'll get based purely on the math, but that's to be expected. Joining a team that will have an easier path to the finals (for example, Riga this season) will definitely give you a better chance at winning if they are strictly equal to the teams in the stacked conference (for which I don't think Riga is a good example, but they did end up with the same number of points in the standings as New York, so whatever).

 

Odds of winning: 50% (1/2)

 

Scenario 2: Two players in strong conference

Here's where things begin to get a little muddled. Now I have to take into consideration that one of the playoff teams will receive a bye (in the weak conference it didn't matter because of the assumption that the third playoff team would be irrelevant over there). Ultimately what I have to do is just figure out the odds of one of your players making it to the finals, and then cut that in half because it'll be a 50/50 against whoever is representing the other conference, a team which you will not have a player on.

 

To begin, it's a 2/3 chance that you'll have the bye, since you are on two teams.  If you have the bye, there is a 1/2 chance that you end up with your second team also in the semi-finals. If that happens, you're in the finals. There's also a 1/2 chance that you have to face the other team, at which point you have a 1/2 chance of winning and advancing to the finals. So overall, if one of your teams has the bye, it's a 75% chance of advancing to the finals (because 1/2 + 1/2 * 1/2).

 

There's a 1/3 chance that you won't have the bye. If that happens, then there's simply a 50% chance that you'll make the finals, because one of your teams advances past round one and from there it's a 1/2 chance.

 

So now if we add up having the bye and not having the bye, it's 2/3 * 75% + 1/3 * 50% = 66.67% (or 2/3). There's a 2/3 chance of you making it to the finals, at which point you'll have a 1/2 chance of winning.

 

Odds of winning: 33.33% (1/3)

 

Scenario 3: One player in each conference

In this scenario, I think it can be a little tricky to look at it from the perspective that if both of your teams make it to the finals, then you win a cup no matter what. Instead, we'll just look at each of your teams chances of winning, and then add them together. Theoretically speaking, I would assume that you have a 1/3 chance of reaching the finals from the strong conference and a 1/2 chance of reaching the finals from the weak conference. I mean, why would it be anything different? There's no reason. That is exactly correct (again, following the assumption that each team has an equal shot against each other team).

 

The strong conference team will have, overall, a 1/6 chance of winning the cup. The weak conference team will have a 1/4 chance. Add those together, and you get a 5/12 chance of winning, or 41.67%.

 

Odds of winning: 41.67% (5/12)

 

 

Final Conclusion

Of course having a player on each of two teams in a conference where only two teams are competing will give you the best odds. Straight away you have a guaranteed place in the finals. However, being on two teams in the same conference is worse than being on teams in different conferences if it's the strong conference. I guess in other words you could just say the more teams you are on in the weak conference, the better your chances of winning. This is, again, assuming we have five top teams who match up exactly equally, which is not a situation that is going to happen in reality.

 

The question the prompted this math did not call for a separation of odds between 'strong' and 'weak' conferences, however, just the chances of winning being in the same or different conferences. Scenario 1 and Scenario 2 need to have their odds combined, and I am going to assume that each has a 50% chance of occurring. So ultimately, if you are on two competing teams in the same conference, you have a 1/4 chance of winning via Scenario 1 and a 1/6 chance of winning via Scenario 2. Add those together and you get exactly the same odds of winning via Scenario 3: 41.67% (5/12).

 

So as you could probably expect, mathematically speaking, if there are five equal teams competing for the Cup, there is no advantage regarding which conferences your players on in with the only notable rule being that the more teams you are on in the weak conference, the better your chances are, but that rule would obviously be the same if you have just one player. Now some of you are probably going to read this and think 'duh, of course it doesn't matter,' and to that I say whatever, it gave me something to write about.

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