Equation 2: 
50-50 and Even-Money Equivalent Bets
by
Alan Krigman
ICON Inc
211 S 45th St
Philadelphia PA 19104
e-mail:
alan@icon-info.com
In Chapter 9, Appendix C, of The Theory of
Blackjack, and in Chapter 9 of Extra Stuff: Gambling
Ramblings, Peter Griffin presents the idea of an
"even-money equivalent bet" which yields the same
expectation and variance as its actual prototype. Griffin uses
the concept as a means of overcoming the limitations of a
classical equation for the probability of being ruined before
achieving a specified win goal. The applicable risk of ruin
equation is given without proof, as it is in Epstein's The
Theory of Blackjack. The same result is derived by Richard
Reid in a paper published at the Math of Blackjack web site
(http://www.bjmath.com/).
The even-money equivalent, and a related 50-50 equivalent, are
useful artifacts independent of risk of ruin analyses. In
particular, they can serve as standards for comparing wholly
different bets and betting progressions because
they capture both the edge and volatility of the wagers in a
manner with a high degree of intuitive appeal.
An even-money equivalent is a hypothetical bet of a derived size,
paying 1-to-1, and its associated probability of winning, that
yields the same expectation and variance as a prototype wager.
A 50-50 equivalent is a hypothetical bet of a derived
size, and the magnitude of its associated payoff, that has a 50%
probability of winning or losing, and yields the same expectation
and variance as a prototype wager.
The principal statistical limitation of the equivalent bet
concept is that it loses the information associated with
"higher" moments of the prototype distributions. That
is, equivalent bets account for expectation (first moment) and
variance (second moment), but not for skewness (third moment),
kurtosis (fourth moment), and so forth. Skewness is relevant to
analysis of the short-terms typical of most gambling sessions, if
not careers. Although skewness explains why many
"systems" seem to work, as well as why bettors
lose so often in games with low-probability high-jackpot payouts
even when they have theoretical returns close to 100%, this
parameter is rarely considered. Kurtosis and higher moments
appear to be of little interest -- although this observation may
result more from a lack of understanding than of fact .
Fortunately, in many gambling situations, approximations are
enough for the intended purposes with only expectation and
variance taken into account. Games with short odds and small
edge, positive or negative, are in this category.
The equations for even-money and 50-50 equivalent bets are relatively straightforward to derive. Assume the expectation and variance of the prototype wager are given as E and V, respectively. B is the amount of the equivalent bet, W is the amount collected when the proposition wins, and P is the probability of a win. Expressions relating the known expectation and variance to the unknown bets, wins, and probabilities are given as Equations 1 and 2.
Equation 1:
P*W + (1-P)*(-B) = E
Equation 2:
For the even-money equivalent, set W=B in Equation 1 and solve to get Equation 3.
Equation 3:
Substitute Equation 3 into Equation 2. After a reasonable amount of cumbersome but straightforward algebra, this reduces to the quadratic in P shown in Equation 4.
Equation 4:
4p2 - 4p + (V/(E2 + V)) = 0
Equation 4 is of the "standard" form ax 2 + bx + c = 0, whose solution is x = [-b +/- sqrt(b 2 - 4ac)]/2a.
Applying this solution yields Equation 5 for probability of winning the even-money equivalent bet as a function only of the expectation and variance of the prototype. A little logic indicates that the positive sign yields the correct root because the probability will be greater than 0.5 when expectation is positive and less than 0.5 when expectation is negative.
Equation 5:
Plugging Equation 5 with the positive sign back into Equation 3 then yields the magnitude of the even-money equivalent bet as a function only of expectation and variance. The final outcomes are shown together as Equations 6 and 7. These forms differ slightly from those in Griffin's books; Griffin used the average squared result of the bet to simplify the calculation.
Equation 6:
Equation 7:
For the 50-50 equivalent, set P=0.5 in Equations 1 and 2 to get Equations 8 and 9.
Equation 8:
Equation 9:
Solve Equation 8 to get B = W - 2*E and plug the result into Equation 9. This leads directly to Equations 10 and 11 for W and B, respectively, as functions only of E and V.
Equation 10:
Equation 11:
These equations are relatively simple to evaluate, to find the
even-money and 50-50 equivalents of any bets for which
expectation and variance are known. An Excel-97 spreadsheet
accompanies this paper (equiv_bets.xls), which may be downloaded for a rapid analysis of various wagers.
Use of the even-money equivalents in analyzing the chances of
achieving various win levels before exhausting a bankroll will be
discussed in a subsequent paper. For present purposes,
application of the equivalents focuses on offering a means of
comparing otherwise disparate bets.
A $10 bet on the pass line with double odds has an expectation
of-$0.1414 and a variance of 816.75. The even-money equivalent is
a bet of $28.579 with a 49.753% chance of winning. The 50-50
equivalent is a bet of $28.720 to win $28.437.
One illustration of the way to use the above figures for
comparison is to see the types of sessions a $10 flat bettor at
blackjack might experience, relative to a person who plays
single-zero roulette with $10 on black every spin. For the
blackjack player, the game is like betting $12.25 with a 49.78%
chance of winning even money. For the $10 roulette player, the
game is like betting $10 with a 48.65% chance of winning even
money. Clearly, the roulette player can expect smaller bankroll
fluctuations during a session, and has less chance of success.
Likewise, a player accustomed to the bankroll swings experienced during a session of roulette betting $10 on the outside will be in for a shock at a Caribbean Stud table with a $10 ante. The former is like flipping a coin for $10.26 to win $9.72. The latter is like flipping the same coin for $22.98 to win $21.92. If the person is comfortable with the sessions experienced playing roulette this way for $10 a shot, these figures suggest that Caribbean Stud with a $5 ante -- equivalent to a coin toss for $11.49 to win $10.96 -- would be preferable.
