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**.

**Expectation --** **
Symbol:
E**.

**Line --** **
Symbol: L**.

**Chance --**
*
Symbols: *

**Odds --** *
Symbols: *

**Residual --**
**
Symbol: R**.

**Cushion --**
**
Symbol: Z**.

**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)

© Dave Zes 2004

** **