In another thread I noted that a player who is able to consistently pick winners with better than 52.38% frequency can increase his expected return (and assumed risk) by betting parlays. I also referred to the fact that even a player who is unable to pick winners at better than 52.38% couldtheoreticallyincrease his expected return by bettingcorrelatedparlays.

Now before you coin-flippers get your hopes up, let me state that in practice you probably can’t expect to find any correlations sufficiently strong so as to overcome the inherent disadvantage of blindly betting parlays. In fact, in the situations where the strongest correlations do exist, most books won’t even allow you to parlay such bets. Nevertheless, understanding the value of parlaying even slightly correlated bets can be instructive.

One might well ask just what exactly correlation is. Well, Webster defines it as, “a relation existing between phenomena which tend to vary, be associated, or occur together in a way not expected on the basis of chance alone.” In other words, two events are correlated if when one event occurs, the other event is more likely to occur. For example there’s a correlation between smoking and lung cancer.

One rather silly example of correlation exists when flipping a coin. One can say that a coin landing heads ispositively correlatedwith tails being on the bottom side of the coin. There is positive correlation because when one event occurs (the coin landing heads) the other event (tails on the bottom) is more likely to occur than usual (negative correlation would mean less likely than usual -- the usual being 50%). Because when the coin lands heads there’s actually a 100% chance (neglecting the possibility of a phony coin) that the bottom side is tails, this is known asperfect correlation. Of course this is silly because as we all know that heads and tails are just different sides of the same coin and so by course one must always imply the other. Still this is illustrative of a degenerate case of correlation.

So let’s look at the math behind parlaying two bets with perfect correlation at 13/5 versus betting each one straight at -110. Well, if we bet each bet straight at -110, then we know that our expected return is -4.545%. However, due to the perfect positive correlation we know that with 50% likelihood we’ll win both bets and with 50% likelihood we’ll lose both bets. Therefore, if we parlay the two then our expected return:Now admittedly, this is the most extreme possible example of a correlated 2-team parlay and of course one would never find a book willing to allow the parlay of two perfectly correlated events (for instance one could never parlay Team A winningCode:= 50% x 13/5 + 50% x -1 = 80%andTeam A’s opponent losing). However, I hope that this simple example shows that with a correlated parlay it’s possible to turn two negative expectation bets into one positive expectation bet.

So let’s step it up a bit and look at a (very) slightly more realistic example of a pair of correlated bets. Imagine a hypothetical college football game between a very strong team and a very weak team. Let’s say that the very strong team is favored by 41 ½ points and the total is 45 ½ points. Now it’s my claim that a correlation exists between the very strong team covering and the total going over. This is because for the strong team to cover we know that at least 42 points must be scored and if that occurs then anything more than a field goal past that and the total will go over as well. Hence, there’s a very small range of possible scores where the strong team can cover AND the under can hit (only 42-0, 43-0, 44-0, 45-0, 42-2, 43-2, or 42-3). Now note, that while the chance of the strong team covering is only 50% and the chance of hitting the over is also only 50%, the chance of hitting the overgiventhat the strong team has covered is almost certainly considerably higher than 50%. Hence, this situation would be likely to present a profitable opportunity.

Now a total of 45 ½ and side of 41 ½ clearly present a highly unrealistic circumstance to say the least, but nevertheless for the same reasons such a correlation also holds in more typical situations (although to a lesser extent). Ceteris paribus, when dealing with spreads one can expect some degree of correlation between the favorite covering and the total going over and also between the dog covering and the total going under.

Because the spreads are typically such a large percentage of the total, this is especially true with hockey puck lines and baseball run lines. While it’s typically difficult to find a book which will allow you to parlay run/puck lines with totals (and of course no book would let you parlay a run/puck line with a money line – the correlation on that would simply be way too high), such books nevertheless do exist.

One other structural example of correlation exists in baseball between the money line and the total. Because if the home team is winning after 8 ½ innings the game ends, there is less of an opportunity for the game to go over. Hence, there tends to be a correlation between the home team covering and the under and conversely between the away team covering and the over. This correlation is not very large (batting in the bottom of the ninth tends to add less than ¼ of a run on average), but it is nevertheless undeniably present and the parlay opportunity is readily available at most books. Now this doesn’t mean that one should go out and blindly bet this, but if an advantaged bettor tends to like the over and the away team anyway (or the under and the home team) a parlay can make these bets extra attractive.

There’s one last example of correlation I’d like to briefly mention. Successfully handicapping a game is typically about making predictions as to how each team is likely to perform. Now if one is able to isolate those factors that are most likely to affect the outcome and use the exact same set of predictions about those exact same factors to predict the winner ATSandthe total, then one is likely to have a correlation. This is true insofar as if that prediction set turns out accurately, both bets become more likely to win. For example, let’s assume that one feels a particular quarterback to be vastly overrated and let’s further assume that one believes this is by far the most important factor in determining both the total and the winner ATS. Now if this prediction turn out to be true, then it’s more likely that the game will go under (less production than expected by the odds makers from the quarterback) and also that the team with the underrated quarterback will lose (again less production from the quarterback than expected). In other words a correlation exists.

Now this isn’t to say that one can take any harebrained set of predictions one likes and use them to impute correlation, but if one often find that when one’s predictions prove true both the total and the spread (or the total and the money line) tend to win, then it certainly may make sense to explore using parlays to leverage larger returns in these types of situations.

In short, betting a correlated parlay consisting of two events of given likelihood clearly gives one an advantage over betting an uncorrelated parlay consisting of two events of similar likelihood. Correlation, however, is not a panacea. This is to no small extent due to the fact that the most highly correlated events are simply not parlayable. Hence, its not recommended for an unadvantaged player to seek correlated bets to parlay. Even with the correlation he’s not likely to decrease his disadvantage to even the level of a straight bet (although his disadvantage would certainly be less than for an uncorrelated parlay). That being said, if anadvantagedbettor tends to like two slightly correlated bets anyway, then there’s even more reason to parlay them than if the bets were independent. The added risk is of course always a factor with parlays, but the additional expected return of a correlated parlay might just be enough to change the mind of the advantaged bettor who avoids parlays from a risk managment perspective.