Plackett (1975) Probabilities

[CHADS]

Im looking for an equation that Will Give me the Probability percentage of any horse , in a race of any number of runners , finishing in the top 3 positions of the race . All the participants Chances of winning are known .....

 

Plackett's paper of horserace probability

 

Im not a Mathematician but ive found an equation from Plackett 1975 ..... Which as far as I can make out gives the chance of the highest 3 horses , with greatest chance of winning , that they will finish in that order .

 

Im probably missing the obvious becuase im not a mathmagician but does any one know how to find the chance of a single horse finishing in the top 3 positions where all runners chances of winning are known and for any number of runners ..... :Whistle:

[Pyrotex]

Im looking for an equation that Will Give me the Probability percentage of any horse , in a race of any number of runners , finishing in the top 3 positions of the race . All the participants Chances of winning are known ..... :hyper:

Oddly enough, this problem came up recently.

 

Plackett's first equation in his paper is basically correct, though it contains a subtle assumption which is probably as near correct as math of this kind can get.

 

If the probability of horse n coming in 1st is (where x is known)

 

Pn,1 = x%

 

then we assume that if horse m were removed from the race, the probability of horse n coming in 1st would be

 

(-m)Pn,1 = Pn,1 / ( 1 - Pm,1)

 

Therefore, if horse m IS in the race, then the probability of horse m coming in 1st and horse n coming in 2nd is

 

= Pm,1 * (-m)Pn,1

 

= Pm,1 * Pn,1 / ( 1 - Pm,1)

 

The math for calculating the probability of horse n coming in 2nd out of a number of horses must include the probability of each of the other horses winning. So, if there are 7 other horses then you must add up the odds of horse n coming in 2nd, 7 different ways:

 

Pn,2 = (-a)Pn,1 + (-b)Pn,1 + (-c)Pn,1 + ... + (-g)Pn,1

 

The math for calculating the probability of horse n coming in 3rd out of a number of horses must include the probability of each of the other horses winning AND one of the remaining horses coming in 2nd. So, if there are 7 other horses then you must add up the odds of horse n coming in 3rd, 7 * 6 = 42 different ways.

 

For example, for horse a coming in 1st and horse b coming in 2nd and horse n coming in 3rd, we will have the single beginning term

 

= Pa,1 * (-a)Pb,1 * (-a-b)Pn,1 ...

 

This expands to

 

= Pa,1 * { Pb,1 / ( 1 - Pa,1) } * { Pn,1 / ( 1 - Pa,1 - Pb,1 ) } ...

 

You will need many terms like this, with a and b permutating to all other possibilities. Sum them all up to get Pn,3.

 

Finally, the probability that horse n will come in 1st, 2nd OR 3rd is

 

Pn,123 = Pn,1 + Pn,2 + Pn,3

 

for R+1 horses on the track, you will have to add up

 

1 + R + R*(R-1) terms

 

This is a LOT easier to do in a spreadsheet.

[CHADS]

Brilliant!

Thank you very Much Pyro .

 

I was searching for way of deducing whether to Back a Race Horse to Win Only or be placed Each way(top3).

 

Im using the Kelly Criterion in my staking Plan and was inclined to seek a similar type equation for deciding whether a horse deserved a win Only Bet.

If a horses chance of finishing in the top 3 where really high then a win only bet would be preffered if coupled with whether the returns on investment was small .

Big returns on investment plus above average chance of top 3 finish ,with a certain horse, would merit a place Bet .... 1/4 of Boolmakers odds.

 

Although a Place bet is a saftey bet I dont want to unduly stake twice as much money for saftey when double the amount can be put on win only in when certain criteria are met.

 

Im sure a graph can be sculptured eventually .... :shrug:

[Pyrotex]

Brilliant!

Thank you very Much Pyro .

 

I was searching for way of deducing whether to Back a Race Horse to Win Only or be placed Each way(top3).

 

...Although a Place bet is a saftey bet I dont want to unduly stake twice as much money for saftey when double the amount can be put on win only in when certain criteria are met...

 

You're welcome.

 

It's not "brilliant" actually, just the everyday kinda stuff that a NASA Rocket Scientist / Excel Spreadsheet Wizard / Physicist / Raconteur / All Around Nice Person has to do in the office many times a day to keep the paychecks coming in. :hihi:

 

I actually built a spreadsheet that calculates Pn,2 Pn,3 and Pn,123 for a field of five horses.

 

You might want to know that generally speaking, I find that if horse n has the highest odds of winning, then:

 

Pn,3 < Pn,2 < Pn,1

 

That's right, the horse is MORE likely to come in first than second, and MORE likely to come in second than third. Does that match with your experience? :shrug:

[CHADS]

Well at the Moment if I choose 2 horses then I can get at least 1 in the top 3, 60% of the time over n races. Ive not really kept strict data on the races ive speculated on but im beginning to see the benefit of keeping statistics and using math to minimize waste.

 

Iv'e only just started making my own probability figures for each horse in the race but the probabilities obviously change through the course of the race and many times the certainty LOL is Last :shrug: .

If the factors put into finding a horses winning probability dont succumb to too much variance throughout the course of a race then hopefully my attempt at solving the puzzle Logically will be favourable in the long term ..... and at 20-1 :hihi: excellent value .

[Pyrotex]

Okay.

I know what you're thinking.

You're thinking nobody can build a spreadsheet to do this.

He's just bullshitting.

But if there's a chance... even a teeny chance that he CAN do it,

I sure wish he would do it.

And post the spreadsheet here.

 

But of course, he won't.

 

:shrug:

 

Here it is.

It handles any field from 3 to 20 horses.

Follow the simple instructions.

Enjoy