In answer to this a question may be proposed:--Suppose I toss up a half-penny, and you are to guess whether it will be head or tail--must it not be allowed that you have an equal chance to win as to lose? Or, if I hide a half-penny under a hat, and I know what it is, have you not as good a chance to guess right, as if it were tossed up? My KNOWING IT TO BE HEAD can be no hindrance to you, as long as you have liberty of choosing either head or tail. In spite of this reasoning, there are people who build so much upon their own opinion, that should their favourite horse happen to be beaten, they will have it to be owing to some fraud. The following fact is mentioned as a 'paradox.' It happened at Malden, in Essex, in the year 1738, that three horses (and no more than three) started for a L10 plate, and they were all three distanced the first heat, according to the common rules in horse-racing, without any quibble or equivocation; and the following was the solution:--The first horse ran on the inside of the post; the second wanted weight; and the third fell and broke a fore-leg.(54) (54) Cheany's Horse-racing Book. In horse-racing the expectation of an event is considered as the present value, or worth, of whatsoever sum or thing is depending on the happening of that event. Therefore if the expectation on an event be divided by the value of the thing expected, on the happening of that event, the quotient will be the probability of happening. Example I. Suppose two horses, A and B, to start for L50, and there are even bets on both sides; it is evident that the present value or worth of each of their expectations will be L25, and the probabilities 25/50 or 1/2. For, if they had agreed to divide the prize between them, according as the bets should be at the time of their starting, they would each of them be entitled to L25; but if A had been thought so much superior to B that the bets had been 3 to 2 in his favour, then the real value of A's expectation would have been L30, and that of B's only L20, and their several probabilities 30/50 and 20/50. 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.