Mathematics and Betting Thread

[Doobs]

I have been meaning to put all the maths and betting stuff in one thread for some time.  

It will be a place to discuss optimal mathematical strategy for all the below and try and explain it with some maths.  I'll call it optimal mathematical strategy, but think it could be easily described as optimal strategy for most of us.  I will also try and give rough rules about when these bets become no brainers and others where they should be avoided.

If people ask what they should do with a free bet, we can just direct them here, and hopefully the answer will be easier to find than if it was buried on page 1145 on TfT.

At the start I am going to cover:
free bets;
bets with money back/free bets for second;
bets with money back/free bets for second to the favourite;
the Richard Hughes promotion for Ascot;
dirty each way
best price guarantees

If a bookie offers something that doesn't look like any of the above, we can discuss it here.

[Doobs]

Free Bets

The usual format of a free bet is that you get the correct price on your free bet, but lose the assumed stake money.

I used to fall into the trap of using free bets to back short price outcomes thus locking up some free bet money.  Though I have made hundreds of pounds over the years, I have definitely wasted hundreds more in EV over the years.  So next time somebody says "If I back Barcelona to win and then back them straight away after the match starts then I effectively get Barcelona at evens instead of 1/2, then please point them here.    

There are two basic rules of free bets.  
1) Only use them to back horses/events at big prices.
2) Never use them to back each way.

Say you had a £50 free bet.

1) In this thread I will estimate a chance of winning, the reality will tend to be that I am overstating the chance of winning as Bookies make a profit.  But the estimates are going to be close enough to correct to show why we should always choose a big priced horse/event.  

If you back a 1/2 chance, you can assume that it has a 67% chance of winning.  When we win, we win £25.  Hence your expected return is £25 x 0.67 = £16.67
If you back an evens chance.   Assume it has a 50% chance of winning.  When we win, we win £50, your expected return is 0.5 x £50 = £25
If you back a 2/1 chance.  Assume it has a 33% chance of winning, your expected return is 0.333 x £100 = £33.33
If you back a 3/1 chance.  Assume it has a 25% chance of winning, your expected return is 0.25 x £150 = £37.50
and so on.

If we carry on, then the expected return from backing:
a 5/1 chance is £41.67,
a 7/1 chance is £43.75,
a 10/1 chance is £45.45,
a 14/1 chance is £46.67,
a 20/1 chance is £47.62.

I think once you get past 7s or so, it is probably is a bit flatter than illustrated, as bookies tend to offer genuine 40/1 and 50/1 chances at 33/1.  

So I think the answer is make sure you at least get a decent price on your free bet.  I am sure that if you take a horse from 7s upwards, you aren't losing a lot of EV by using the free bet this way.  

There are exceptions.  Sometimes the free bets are time limited and it is always better to take any bet on than let it expire, hence sometimes I take shorter prices than I would normally like, and on other occasions you may think a 5/1 horse should be 3s, so there isn't a lot of harm done there.

2) why you should never use free bets to back each way.

Assume I back a 4/1 horse each way getting quarter the odds, if it gets a place my return on a normal bet would be my stake back on a normal bet.  Under a free bet I lose my initial stake so my actual return is zero (£50 on a normal bet).  

Likewise, if I took 6/1, I'd only get £25 back (£75 on a normal bet), and if I took 10s, I'd only get £75 back (£125 on a normal bet).

Basically you are killing a lot of EV even backing big priced runners each way.
 

[Doobs]

Money Back Offers if Your Horse is Second.

There seems to be a lot of these offers around at the moment and taking up the offer is usually correct, but there are definite exceptions.  I haven't yet had an account closed that has obviously been as a result of taking up offers, though have had my access to offers removed from an account where I made most of my money without using offers.

I think the golden rule of taking these offers is:

Short is Good

If you have two horses in a race that are 2/1 and 5/1.  Without any extra research I can assume the 2/1 horse is more likely to be first, but not only that I can see that the 2/1 horse is more likely to be 2nd.  Getting the horses that are more likely to be 2nd is where you get the value from. 

So if I have 5 minutes, I can check that the 2/1 price is close to the Betfair price and close to the best price.  If that checks out then I can assume I am not going to lose much value from the bet.  Not only that but because the Betfair market is close to perfect I can assume that backing the 2/1 is going to be the best option way more of the time than the 5/1 chance and conclude I don't add much by doing "research". 

It is always the case that shorter is always better, in the Coronation Cup the other week, I was fairly certain that Dunaden was more likely to be 2nd than first, but that was only because the favourite was so short (2/5).

Maths

Throughout the examples, I will assume that Betfair midprice is a good proxy for the real chance of the horse winning a race.

Assuming that we have a money back offer and are betting on a horse that is 2/1 with a bookie under this offer and mid price 3.25 on Betfair. 
Assuming we get cash back if 2nd.  Our expected return without the cash return from a £50 bet is 50/3.25 x 3 = £46.15. 
If the horse is a 30% chance of finishing first, we can assume the horse is probably around a 25% chance to be 2nd.  So our return from the money back offer is an additional .25 x 50 = £12.50

So total expected return is £46.15 + £12.50 = £58.65 or an ROI of 17%.  If your return is a free bet, if you use it wisely, you probably lose 10% in the long run from the £12.50, so your ROI drops to 15%   

The maths becomes less compelling as the price of our original horse gets bigger and if we move away from the Betfair price.

Assume that we have a money back offer and are betting on a horse that is 8/1 with a bookie under this offer and mid price 10s on Betfair. 
Assuming we get cash back if 2nd.  Our expected return without the cash return from a £50 bet is 50/10 x 9 = £45. 
If the horse is an 11% chance of finishing first, we can assume the horse is probably around an 11% chance to be 2nd.  So our return from the money back offer is an additional .11 x 50 = £5.50

So total expected return is £45 + £5.50 = £50.50 or an ROI of 1%.  If your return is a free bet, if you use it wisely, you probably lose 10% in the long run from the £12.50, so your ROI drops to 0%

Interestingly you get a better ROI backing a 2/1 chance that is a genuine 9/4 chance than you do backing a horse that is 8/1 and should be 8/1.  If you don't know much about the horses, then just assume short is best.   

[Doobs]

Money Back Offers if Your Horse is Second to a Named Horse.

These offers aren't nearly as generous as the offers that give a return if your horse is 2nd to any horse.  I'd say it is easy to lose money using these offers if you aren't careful 

I think the golden rule of taking these offers is:

Short prices on both horses are essential

If you back a horse at 6/1 and the offer is a return of cash if your horse is 2nd to another 6/1 chance, then the chance of them finishing one to two in the correct order is going to be something close to 2%.  If the bookie then offers you worse odds than Betfair that aren't even best odds then it is an easy offer to turn down this offer.  I was sent an offer very close to this early on Saturday (I actually had to pick 2nd to an unnamed favourite, so it was even worse).

To take up an offer like this it is very important that both horses are short.

Maths

Throughout the examples, I will assume that Betfair midprice is a good proxy for the real chance of the horse winning a race.

Assume we are offered money back if our 5/1 horse is 2nd to an evens favourite

Assuming that we have a money back offer and are betting on a horse that is 5/1 with a bookie under this offer and mid price 6.5 on Betfair. 
Assuming we get cash back if 2nd.  Our expected return without the cash return from a £50 bet is 50/6.5 x 6= £46.15. 
If the horse is a 50% chance of the evens horse finishing first, and we can assume the horse is probably around a 30% chance to be 2nd if the evens horse wins.  So our return from the money back offer is an additional .5 x.3 x 50 = £7.50.  So our return is £53.65 or an ROI of 7%.  If your return is a free bet, if you use it wisely, you probably lose 10% in the long run from the £7.50, so your ROI drops to 6%   

The maths becomes less compelling as the price of each of the horses gets bigger and if we move away from the Betfair price. 

Your probably struggling to make a positive return once the horse we have to finish 2nd to gets much above 2/1 unless the Bookie is right on Betfair prices for the horse we back or we are picking our horse with an edge.  If you see this offer in a big handicap, it should be pretty easy to just bin it without much investigation.

[Doobs]

Money back if a named horse wins

These bets have the same additional value whichever horse you back (barring the named horse).

So if you are backing blind, you want to get the horse that is best value vs Betfair.

You can approximate the chance of a horse winning by taking betfair mid price.  So if it is available to back at 5.4 and lay at 5.5, the mid price is 5.45.  If your bookie offers you 4/1 on this horse (which equates to 5 in decimal odds) you are getting back 5/5.45 = 91.7% of the value of the bet from taking the bad horse.  So long as the refund you get from the named horse is worth more than 8.3%, then this bet is +EV.

Example
There is an example at Royal Ascot where Paddy Power are giving you money back if Animal Kingdom wins the Queen Anne.  Given Animal Kingdom is odds on, you get just over £25 (or 51.6%)  of value added to your free bet if they allow you to bet £50 (The offer is max £100, I can't get £50 on).  This enables you to take much lower prices on Paddy Power than you would do elsewhere, and in addition they have best price guaranteed on the race, so you aren't going to get a bad price no matter what they are quoting now.

On the prices they quote right now the best value hoses against Betfair are
Penitent, Gregorian, Declaration of War,  Gabriel, Chil The Kite, Aljamaheer, Trumpet Major, Sovereign Debt, Elusive Kate and Trade Storm, in that order.  

I think you can assume that those higher up the betting are going to start closer to best price than those at the bottom, so I probably wouldn't choose Penitent or Chil The Kite.

Whatever you do you should definitely take the Paddy Power bet if you can as the added value is so good.

There was a similar offer on the World snooker where you got money back if O'Sullivan won.  The maths in that case were much closer as O'Sullivan was about 6/1.  In that case, I could only find one or two players worth backing.  This is because if the named horse is 6/1 the bookie prices have to be much closer to the true (or Betfair) odds in order to be +EV.

[Doobs]

Money back if a named jockey places

The value of this offer is roughly 90% of the value of 1/(probability of placing of the named jockey) x bet size x probability of your choice losing.

So the offer is
1) more valuable the shorter the price of the Jockey's rides,
2) more valuable the bigger the price of your choice of horse.

Needless to say, you want to pick a horse that is near the Betfair midprice.

Example

3.45 Ascot tomorrow

Probability Toronado places is 1/1.7.  The value of the free bet before adjustment for the probability our horse lose is 25/1.7 = £11.91

Assuming betfair mid price reflects true odds
Mars is the best bet, value of Mars bet alone = 25 x 13(BetVictor's price)/14.75(Betfair Mid) + (1-1/14.75) x 11.91
= £33.03

The next best mathematical bets are Dawn Approach and Magician.
 

[Doobs]

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[Doobs]

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[The Camel]

Brilliant thread, thanks for doing this Doobs.

[Marky147]

Brilliant thread, thanks for doing this Doobs.

Definitely this!

Looking forward to plenty of nosebleeds as I try to follow.

[TightEnd]

Brilliant thread, thanks for doing this Doobs.

Exactly. Thanks very much, very useful

[outragous76]

I assume we can ask questions?

Free bet

Using your theory, is there a point of diminishing return? (Ill assume not, just ever decreasing increase?)

Therefore, although your point is excellent, there must be a point where we say a return is better than bigger ev?

Hence, and where my assumptions lead to, is there a sweetspot price we should aim for?

[The Camel]

I'd be interested on the mathematics of bad each way races.

How much ev on the win part of the bet can we afford to give up when the place part of the bet is clearly +ev?

I do it by intuition only at the moment.

[doubleup]

I'd be interested on the mathematics of bad each way races.

How much ev on the win part of the bet can we afford to give up when the place part of the bet is clearly +ev?

I do it by intuition only at the moment.

Here's a calculator

https://docs.google.com/file/d/0Byg8Wbbo5p2ucDN2WVdueENUYzA/edit?usp=sharing

you have to download it - you can't put the data in on google docs

[The Camel]

I'd be interested on the mathematics of bad each way races.

How much ev on the win part of the bet can we afford to give up when the place part of the bet is clearly +ev?

I do it by intuition only at the moment.

Here's a calculator

https://docs.google.com/file/d/0Byg8Wbbo5p2uN3dRYlQwaTlhNE0/edit?usp=sharing

you have to download it - you can't put the data in on google docs

Great stuff, thanks.