"If the margin at quarter time is a multiple of 6 points I'll pay you $5; if it's not, you pay me a $1. If the two teams are level at quarter-time it's a wash and neither of us pay the other anything.
Now quarter-time margins are unpredictable, so the probability of the margin being a multiple of 6 is 1-in-6, so my offering you odds of 5/1 makes it a fair bet, right? Actually, since goals are worth six points, you've probably got the better of the deal, since you'll collect if both teams kick the same number of behinds in the quarter.
Deal?"
At first glance this bet might look reasonable, but it isn't. I'll take you through the mechanics of why, and suggest a few even more lucrative variations.
Firstly, taking out the drawn quarter scenario is important. Since zero is divisible by 6 - actually, it's divisible by everything but itself - this result would otherwise be a loser for the bet proposer. Historically, about 2.4% of games have been locked up at the end of the 1st quarter, so you want those games off the table.
You could take the high moral ground on removing the zero case too, because your probability argument implicitly assumes that you're ignoring zeroes. If you're claiming that the chances of a randomly selected number being divisible by 6 is 1-in-6 then it's as if you're saying something like the following:
"Consider all the possible margins of 12 goals or less at quarter time. Now twelve of those margins - 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66 and 72 - are divisible by 6, and the other 60, excluding 0, are not. So the chances of the margin being divisible by 6 are 12-in-72 or 1-in-6."
In running that line, though, I'm making two more implicit assumptions, one fairly obvious and the other more subtle.
The obvious assumption I'm making is that every margin is equally likely. Demonstrably, it's not. Smaller margins are almost universally more frequent than larger margins. Because of this, the proportion of games with margins of 1 to 5 points is more than 5 times larger than the proportion of games with margins of exactly 6 points, the proportion of games with margins of 7 to 11 points is more than 5 times larger than the proportion of games with margins of exactly 12 points, and so on. It's this factor that, primarily, makes the bet profitable.
The tendency for higher margins to be less frequent is strong, but it's not inviolate. For example, historically more games have had a 5-point margin at quarter time than a 4-point margin, and more have had an 11-point margin than a 10-point margin. Nonetheless, overall, the declining tendency has been strong enough for the proposition bet to be profitable as I've described it.
Here is a chart of the frequency distribution of margins at the end of the 1st quarter.
The far-less obvious assumption in my earlier explanation of the fairness of the bet is that the bet proposer will have exactly five-sixths of the margins in his or her favour; he or she will almost certainly have more than this, albeit only slightly more.
This is because there'll be a highest margin and that highest margin is more likely not to be divisible by 6 than it is to be divisible by 6. The simple reason for this is, as we've already noted, that only one-sixth of all numbers are divisible by six.
So if, for example, the highest margin witnessed at quarter-time is 71 points (which, actually, it is), then the bet proposer has 60 margins in his or her favour and the bet acceptor has only 11. That's 5 more margins in the proposer's favour than the 5/1 odds require, even if every margin was equally likely.
The only way for the ratio of margins in favour of the proposer to those in favour of the acceptor to be exactly 5-to-1 would be for the highest margin to be an exact multiple of 6. In all other cases, the bet proposer has an additional edge (though to be fair it's a very, very small one - about 0.02%).
So why did I choose to settle the bet at the end of the 1st quarter and not instead, say, at the end of the game?
Well, as a game progresses the average margin tends to increase and that reduces the steepness of the decline in frequency with increasing margin size.
Here's the frequency distribution of margins as at game's end.
(As well as the shallower decline in frequencies, note how much less prominent the 1-point game is in this chart compared to the previous one. Games that are 1-point affairs are good for the bet proposer.)
The slower rate of decline when using 4th-quarter rather than 1st-quarter margins makes the wager more susceptible to transient stochastic fluctuations - or what most normal people would call 'bad luck' - so much so that the wager would have been unprofitable in just over 30% of the 114 seasons from 1897 to 2010, including a horror run of 8 losing seasons in 13 starting in 1956 and ending in 1968.
Across all 114 seasons taken as a whole though it would also have been profitable. If you take my proposition bet as originally stated and assume that you'd found a well-funded, if a little slow and by now aged, footballing friend who'd taken this bet since the first game in the first round of 1897, you'd have made about 12c per game from him or her on average. You'd have paid out the $5 about 14.7% of the time and collected the $1 the other 85.3% of the time.
Alternatively, if you'd made the same wager but on the basis of the final margin, and not the margin at quarter-time, then you'd have made only 7.7c per game, having paid out 15.4% of the time and collected the other 84.6% of the time.
One way that you could increase your rate of return, whether you choose the 1st- or 4th-quarter margin as the basis for determining the winner, would be to choose a divisor higher than 6. So, for example, you could offer to pay $9 if the margin at quarter-time was divisible by 10 and collect $1 if it wasn't. By choosing a higher divisor you virtually ensure that there'll be sufficient decline in the frequencies that your wager will be profitable.
In this last table I've provided the empirical data for the profitability of every divisor between 2 and 20. For a divisor of N the bet is that you'll pay $N-1 if the margin is divisible by N and you'll receive $1 if it isn't. The left column shows the profit if you'd settled the bet at quarter-time, and the right column if you'd settled it all full-time.
As the divisor gets larger, the proposer benefits from the near-certainty that the frequency of an exactly-divisible margin will be smaller than what's required for profitability; he or she also benefits more from the "extra margins" effect since there are likely to be more of them and, for the situation where the bet is being settled at quarter-time, these extra margins are more likely to include a significant number of games.
Consider, for example, the bet for a divisor of 20. For that wager, even if the proportion of games ending the quarter with margins of 20, 40 or 60 points is about one-twentieth the total proportion ending with a margin of 60 points or less, the bet proposer has all the margins from 61 to 71 points in his or her favour. That, as it turns out, is about another 11 games, or almost 0.1%. Every little bit helps.
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