Programación botones de registro: Zona I (Captura de Registros)

There are three main ways that a punter can bet on the results of a race — in a betting shop either locally or at the track, through a bookmaker at the track, or by the Tote. The Tote is a system which, by its very nature, cannot lose money. At the time a punter places a

£10 bet on “My Fair Lady” with the Tote he has only a rough idea of what he would win if that horse won. The Tote organizers will take the total money staked on the race, say £500,000, take a proportion of this, say £25,000, to cover expenses and to create a profit and will divide the remaining money, £475,000, between those that backed

‘My Fair Lady’ to win in proportion to how much they staked. Thus, if the people backing “My Fair Lady” staked a total of £20,000 on the horse then our punter would receive a fraction 10/20,000 of the returned stake money. Thus his return will be

£475, 000× 10

20, 000 = £237.50.

The Tote is actually a little more complicated than we have described because “place bets” can be made where the punter will get a reduced return, a fraction of the odds for winning, if the horse finishes in one of the first two, three, or four places, depending on the number of horses in the race. However, the essentials of the Tote system are described by just considering bets to win.

A betting shop or a bookmaker, by contrast, enters into a con-tract with the punter to pay particular odds at the time the bet is placed. If the horse backed has odds 3/1 (three-to-one) then a successful £10 bet will bring a return of the original £10 plus 3 times £10, i.e., £40 in all. A bookmaker can lose, and lose heavily,

on a particular race but if he knows his craft and assesses odds skill-fully he will make a profit over the long term.

Let us take an example of odds set by a bookmaker that would obviously be foolish. Consider a two-horse race where the book-maker sets odds of 2/1 on each horse. A punter would not take long to work out that if he staked £1,000 on each horse his total stake on the race would be £2,000 but, whichever horse won, his return from the race would be £3,000. This case is easy to see without analyzing it in detail but more subtle examples of bad odds-setting can occur.

Let us take a hypothetical race with 10 runners and the bookmaker, assessing the relative merits of the various runners, offers the fol-lowing odds on a race for fillies:

Diana 3/1

Dawn Lady 6/1 Fairy Princess 6/1 Olive Green 10/1

Mayfly 10/1

Dawn Chorus 15/1

Missy 15/1

Lovelorn 20/1

Helen of Troy 25/1 Piece of Cake 30/1

Although it is not obvious from a casual inspection of the list of odds this bookmaker is heading for certain ruin and any punter worth his salt could make a profit on this race. Let us see how he does this. He bets on every horse and places bets as follows:

Diana £250 (£1,000)

Dawn Lady £143 (£1,001) Fairy Princess £143 (£1,001) Olive Green £91 (£1,001)

Mayfly £91 (£1,001)

Dawn Chorus £63 (£1,008)

Missy £63 (£1,008)

Lovelorn £48 (£1,008) Helen of Troy £39 (£1,014) Piece of Cake £33 (£1,023)

In parentheses, after the bets, are shown what the punter will receive if that particular horse wins — anything between £1,000 and £1,023.

However, if the bets are added they come to £964; the punter is a certain winner and he will win between £36 and £59 depending on which horse wins the race. Clearly no bookmaker would offer such odds on a race and we shall now see what the principles are for setting the odds that ensure a high probability of profit for the bookmaker in the longer run.

We notice that what the punter has done is to set his stake on each horse at a whole number of pounds that will give a return (stake money plus winnings) of £1,000 or a small amount more. For simplic-ity in analyzing the situation, in what follows we shall assume that he adjusts his stake to receive exactly £1,000 — although bookmak-ers do not accept stakes involving pennies and fractions of pennies.

If the horse has odds of n/1 then the punter will receive n+ 1 times his stake money so to get a return of £1,000 the amount staked is

£1,000/(n+ 1). To check this, we see that for Diana, with n = 3, the stake is £1,000/(3+ 1) = £250. Now let us take a general case where the odds for a ten-horse race are indicated as n₁/1, n₂/1· · ·n10/1.

If the punter backs every horse in the race, planning to get £1,000 returned no matter which horse wins, then his total stake, in pounds, is

S= 1000

n₁+ 1+ 1000

n₂+ 1+ · · · + 1000

n₁₀+ 1. (3.1) If S is less than £1,000 then the punter is bound to win. Thus for the bookmaker not to be certain to lose money to the knowledge-able punter, S must be greater than 1,000. This condition gives the

bookmaker’s golden rule which we will now express in a mathe-matical form.

If we take Eq. (3.1) and divide both sides by 1,000 then we have S

1000 = 1

n₁+ 1+ 1

n₂+ 1+ · · · + 1

n₁₀+ 1, (3.2) and for the bookmaker not to lose to the clever punter the left-hand side must be greater than 1. Put into a mathematical notation, which we first give and then explain, the golden-rule condition is

m i=1

1

ni+ 1 >1. (3.3)

The symbol > means “greater than” and Eq. (3.3) is the condition that a punter cannot guarantee to win — although of course he may win just by putting a single bet on the winning horse. The summation symbol_m

i=1means that you are going to add together m quantities with the value of i going from 1 to m. In our specific case m = 10, the number of horses, and i runs from 1 to 10 so that

10

If relationship (3.3) were not true and the summation was less than 1, as in the hypothetical case we first considered, then the punter could guarantee to win. The skill of a bookmaker is not just in fixing his odds to satisfy the golden rule — anyone with modest mathematical skills could do that. He must also properly assess the likelihood of each horse winning the race. If he fixed the odds on a horse at 10/1 when the horse actually has a one-in-five chance of winning then astute professional gamblers would soon pick this up and take advantage of it.