Statistics of Gambling: Units of Likelihood

 

Despite the growing popularity of gaming, its math remains poorly documented. This must be owed in part to the fact that many players prefer to handicap qualitatively, so hard numbers are not important. As computational methods become more accessible, quantitative handicapping is sure to gain popularity.

The “Line,” the most common unit in wagering, is not a good mathematical object. Functions that convert other measures of likelihood, such as Probability, to Line are, to use a mathematical term, “discontinuous.” An alternate unit, “Cent Line Residual,” or just “Residual,” derives directly from the line, and can act as a good middleman between conversions. We also introduce “Cushion,” which is a function of two variables: the probability the handicapper has assigned to some outcome, and the Residual payout offered for the success of that outcome.

 

The Units

 

Probability -- Symbol: P. Units: unitless. Definition: Average number of successful occurrences divided by the average number of total occurrences if sampled an infinite number of times. Appearance: A number, as a decimal or fraction, that must be between zero and one. Examples: 0.6250, 5/8.

Expectation -- Symbol: E. Units: unitless. Definition: Average number of net successes (number of successes minus number of losses) divided by the average number of total occurrences if sampled an infinite number of times. Appearance: A number, as a decimal or fraction, that must be between negative one and one. Examples: -0.3333, 2/5.

Line -- Symbol: L. Units: money. Definition: If negative, the amount in dollars credited to wagerer to win $100 (a negative credit is a debt); if positive, the net amount in dollars credited to wagerer for a success for a stake of $100. Appearance: Usually a whole number. Either less than negative one-hundred or greater than or equal to positive one-hundred. Examples: -120, 200, -380, 102.

Chance -- Symbols: S, N. Units: Unitless. Definition: Same as Probability. Appearance: Typically whole positive numbers -- separated by "in". Examples: 5 in 100, 2 in 3, 1 in a million.

Odds -- Symbols: F, S. Units: Unitless. Definition: Average number of failures compared to average number of successes if sampled an infinite number of times. Appearance: Typically whole positive numbers -- separated by "to". Examples: 5 to 95, 2 to 1, 1 to 999,999.

Residual -- Symbol: R. Units: money. Definition: If negative, Line plus one-hundred; if positive, Line minus one-hundred. Appearance: Usually a whole number. Examples: -20, 100, -280, 2. Note: Only used here.

Cushion -- Symbol: Z. Units: money. Definition: The ideal difference, relative to a risk of $100, in dollars, between two differing Lines auguring the same event. Appearance: A number, sometimes whole, or sometimes carried out to a couple of decimal places for accuracy. Examples: -20, 18.5, -3.82, 34.

 

Conversions

 

Probability from Expectation:

Example: If Expectation is 0, then P = ( 0 + 1 ) / 2 = 1 / 2 = 0.5. (graph)

 

Probability from Line:

Favorite Dog

Example: If Line is -180, this is a favorite, so P = -180 / ( -180 - 100 ) = -180 / -280 = 0.6429... If Line is 115, this is an underdog, so P = 100 / ( 115 + 100 ) = 0.4651... (graph)

 

Probability from Chance:

Example: If statement is 5 in 16, then S = 5, N = 16, so P = 5 / 16 = 0.3125.

 

Probability from Odds:

Example: If statement is 5 to 6, then S = 5, F = 6, so P = 5 / ( 6 + 5 ) = 0.4545...

 

Line from Probability:

Favorite Dog

Example: If Probability is 0.100, then this is an underdog, so L = 100 ( 1 - 0.100 ) / 0.100 = 100 * 0.900 / 0.100 = 90 / 0.100 = 900. If Probability is 0.8112, then this occurrence is favored, so L = -100*0.8112 / ( 1 - 0.8112 ) = 81.12 / 0.1888 = 430. (graph)

 

Line from Expectation:

Favorite Dog

Example: If Expectation is -0.100, then this outcome is a slight underdog, so L = 100 * ( 1 - ( -0.100 ) ) ) / ( -0.100 +1 ) = 100 * 1.100 / 0.900 = 122. If Expectation is 0.4500, then this is a favorite, so L = 100 * ( 0.4500 + 1 ) / ( 0.4500 - 1 ) = 100 * 1.4500 / -0.5500 = -264. (graph)

 

Expectation from Probability:

Example: If Probability is 1/2, then E = 2 * ( 1 / 2 ) - 1 = 1 - 1 = 0. (graph)

 

Expectation from Line:

Favorite Dog

Example: If Line is -150, then this is a favorite, so E = ( -150 + 100 ) / ( -150 - 100 ) = -50 / -250 = 50 / 250 = 1 / 5. If Line is 260, then this occurrence is an underdog, so E = ( -260 + 100 ) / ( 260 + 100 ) = -160 / 360 = -4 / 9 = -0.4444... (graph)

 

Chance from Probability:

Example: If Probability is 0.50, then, if we let x = 2, we get S = 2 * 0.50 = 1, and N = 2. So our statement is "1 in 2".

 

Odds from Probability:

Example: If Probability is 1 / 5, then, if we let x = 5, we get S = 5 * 1 / 5 = 1, and F = 5 * ( 1 - 1 / 5 ) = 5 * 4 / 5 = 4. So our statement is "4 to 1".

 

Residual from Probability:

Favorite Dog

Example: If Probability is 9 / 11, then this is a favorite, so R = 100 * ( 2 * 9 / 11 - 1 ) / ( 9 / 11 - 1 ) = 100 * ( 18 / 11 - 1 ) / ( -2 / 11 ) = 100 * ( 7 / 11 ) / ( -2 / 11 ) = 100 * ( -7 / 2 ) = -350. If Probability is 0.49, this is a slight underdog, so R = 100 * ( 1 - 2 * 0.49) / 0.49 = 100 * 0.02 / 0.49 = 4.08... (graph)

 

Residual from Line:

Favorite Dog

Example: If Line is -110, then, as a favorite, R = -110 + 100 = -10. If Line is 220, then R = 220 - 100 = 120. (graph)

 

Cushion from House Line and Optimal Prediction:

Favorite Dog

Same As:

Favorite Dog

Example: We'll use the bottom set of equations. We have two independent variables, RH and P. We use P to decide whether the outcome is a favorite or an underdog. So, suppose our P is 0.625, and the offered Line -160. So RH is -60. So Z = ( 1 - 0.625 ) * ( -60 - 100 * ( 2 * 0.625 - 1 ) / ( 0.625 - 1 ) ) = 0.375 * ( -60 - 100 * ( 1.25 - 1 ) / ( -0.375 ) ) = 0.375 * ( -60 - 100 * ( -2 / 3 ) ) = 0.375 * ( -60 - ( -66.7 ) ) = 0.375 * 6.67 = 2.50. This tells us that it is slightly favorable to accept the house's Line. Now suppose P is 1 / 4, and the house Line is 180. So RH is 80. So Z = ( 1 / 4 ) * ( 80 - 100 * ( 1 - 2 * 1 / 4 ) / ( 1 / 4 ) ) = ( 1 / 4 ) * ( 80 - 100 * 1 / 2 / ( 1 / 4 ) ) = ( 1 / 4 ) * ( 80 - 200 ) = ( 1 / 4 ) * ( -120 ) = -30. This tells us that going against the house is quite unfavorable, that is, that the offered payout of 180 is not enough based on a probability of only 1 / 4. (graph)

 

 

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© Dave Zes 2004