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The Gaming Table by Andrew Steinmetz, Volume II

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The Doctrine Of Probabilities Applied To Gambling

Example II. Let us suppose three horses to start for a
sweepstake, namely, A, B, and C, and that the odds are 8 to 6 A
against B, and 6 to 4 B against C--what are the odds--A against
C, and the field against A? Answer:--2 to 1 A against C, and 10
to 8, or 5 to 4 the field against A. For
A's expectation is 8
B's expectation is 6
C's expectation is 4
----
18

But if the bets had been 7 to 4 A against B; and even money B
against C, then the odds would have been 8 to 7 the field against
A, as shown in the following scheme:--
7 A
4 B
4 C
----
15

But as this is the basis upon which all the rest depends, another
example or two may be required to make it as plain as possible.

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