So like most offices we have a Super Bowl box pool. $20 for five squares (or $5 for one square). We have enough people to fill two boxes.
I am going to partner with someone to put in $40 total between us. Are we better off both in the same box, or in different boxes?
My intuition says that we are "probably" better off with both of us being in the same box, as that gives us 10 chances to win something in that particular box instead of five. (10% chance to win something in one box, 0% chance in other; compared to 5% chance to win in each box).
However, I'm not convinced that the edge is as great as it first appears to be, since it also reduces the chances we could be big winners in both boxes.
Payout is 10% for 1st quarter, 20% for halftime, 10% for 3rd quarter, 60% for final score (nothing for 4th quarter if there is OT). If we assume $400 for each box (everyone puts in $20).
If scores and distribution of boxes is random, hmm....
10% of 400 + 0% of 400 = 5% of 200 + 5% of 200 so I am having trouble reconciling my intuition with math here, I think the math is missing something though since we might have duplication of numbers if spread over 2 boxes...but on a net basis, is duplication bad? (both lose) or good? (both win).
Maybe both of us in one box has better chance of winning something while each of us in separate box has less likely chance but bigger potential payout (shorter wider distribution vs taller narrower distribution with same area under each).
Some days I wish I had gone beyond statistics 101....
PS I always thought it would be fun to have a secondary market after the random selection of boxes. Would getting some combination of 7, 3, 0 be worth more than others?
While I don't understand the specifics of your superb owl pool, it sounds like a variation of the "draw a number, and if it matches [total points, score differential, some condition X], you win" and there are enough participants to have two separate pools, of which you can contribute to one or both.
Your overall odds of winning, assuming all possibilities are equally likely to win, is simply the number of possibilities in your control divided by the total number of possibilities, regardless of how they're broken down.
At its simplest, think of flipping two coins, so there are four potential bets you could make (H1, T1, H2, T2). It costs you $1 to cover one of those. You decide to partner up with someone else to cover two possibilities. You could each take a different side of the same coin, guaranteeing you'll break even, but only break even. One of your bets must lose while the other must win, and you can't win anything from the other coin since you have no money on it.
If instead you each take one side on each coin, you will still be expected to break even. 1/2 the time that will actually happen, 1/4 of the time you'll lose both dollars, and 1/4 of the time you'll come out $2 ahead.
Or another way to think of it is like keno. Your choice is essentially like choosing between playing multiple tickets for a single drawing, or one ticket for multiple drawings. All else being equal, your overall expected value doesn't change.
