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January-01-2009,
The Complete Square's Guide to Sports Wagering: Basic Math for Sports Gamblers...By Jay Graziani

Gamblers are often seen as fearless, throwing hundreds or thousands of dollars on games that are close to 50/50 propositions and whose day-to-day results depend largely on dumb luck. Despite this apparent boldness in the face of risk, many handicappers, especially those of the recreational variety, are terrified of the one thing that is most important to their success: basic math.

Make no mistake about it, gambling is a numbers game. "Handicappers" will spend hours poring over newspaper articles, game footage, and pundits' opinions, yet neglect to understand the simple math that allows them to know what line is the best, estimate their edge, or determine how much to wager. No matter how you look at it, an aversion to numbers keeps money out of your pocket. This article collects some of the essential formulas that every handicapper should have in his toolbox.

Converting American odds to decimal odds
For positive (+) odds: decimal = (moneyline/100) +1
For negative (-) odds: decimal = 1 - (100/moneyline)
For decimal odds of 2 or higher: moneyline = (decimal - 1) *100
For decimal odds less than 2: moneyline = (-100)/(decimal -1)

While most American bettors use (naturally) the American-style odds format, decimal odds are much easier to work with in many situations (for instance, see the section on calculating half point values below).

Converting probabilities to moneylines
Moneyline for favorite = (Pfav)*100/(Pdog)
Pfav = probability of favorite winning
Pdog = probability of underdog winning (or, 1-Pfav)

Knowing the probability of something happening is great, but of little use unless you can convert it to betting odds. Using this formula for a 65% favorite gives us a fair line of (0.65)*100/0.35, or -186. The line for the underdog would just be the opposite, so that a team with a 35% chance of winning should be +186. This formula can also be used to find the winning percentage needed to break even at given odds, by using the given odds and solving for Pfav. For instance, plugging a moneyline of -110 into the formula tells you that you need a 52.4% record to break even against those odds.

Determining half-point "cents" value from push percentage
Odds to buy onto a number= odds - 2*(P%)
Odds must be in decimal format (-110 = 1.91, conversion shown above)
P% = push percentage

If you know the probability of a line "pushing" (ending in a tie), you can easily find the fair odds for buying onto that number. This formula assumes that (1) your push percentage is correct and (2) the currently available line is a "fair" line, i.e. a 50/50 proposition. The push percentage can be derived from historical data. For example, if a team is +6.5 -110 and you think the "7" will push 6% of the time, you can find the fair price for a +7 pointspread. In this case it would be 1.91 - 2*(0.06), or 1.79. Converting back to American style odds gives you a fair value of -7 -127. You should be willing to buy the extra half point as long as it costs 17 "cents" or less, and you are getting value at anything less. Note that the formula is slightly different for buying off a number (e.g. from 7 to 7.5) and is bit complicated to go into here. However, the above formula will provide a close enough estimate for most situations.

The Z-score
Z = (W-L)/sqrt(W+L)
W = number of wins
L = number of losses

The Z-score can be used to determine the statistical significance of a historical record or betting system. Simply divide the net wins (wins minus losses) by the square root of the sample size. A Z-score of 2 indicates a 97% probability that the results are not solely due to chance. A Z-score of 3 means there is a greater than 99% chance that the results of the system are not due to luck alone. For sportsbetting applications, you should look for a minimum Z-score of 3, though even higher is better. More detail on using the Z-score can be found in a previous article (http://www.majorwager.com/index.cfm?page=27&show_column=665).

The Kelly Criterion:
Optimum Risk Percent = W% - [(1-W%)/(W/L)]
W% = win percentage
W = amount "to win"
L = amount "to lose" (risk amount)
For -110 wagers: Optimum Risk Percent = W% - 1.1×(1-W%)

The Kelly Criterion was covered in full in a previous article (http://www.majorwager.com/index.cfm?page=27&show_column=653) and finds the optimum percentage of bankroll you should wager based on your edge and the payout odds. In cases involving -110 odds, the formula can be simplified as shown above, requiring only W% as an input. For a 55% handicapper betting into -110 odds, W% = 0.55, W = 100 and L = 110. Plugging these numbers into the formula gives an optimum bet of 5.5% of bankroll. Keep in mind that "full Kelly" is a very aggressive betting strategy, and betting half- or even quarter-Kelly is generally recommended to avoid excess risk. Also keep in mind that this formula looks at isolated instances. If you have 5 or 10 wagers in play simultaneously, you should scale down your bet size accordingly.

"The Complete Square's Guide to Sports Wagering" is a recurring series aimed at educating novice sports bettors.

01-01-08
Jay Graziani
MajorWager.com
graziani@majorwager.com

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