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March 6th, 2017, 03:08 AM   #1
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The Best (Probability) Combination of Multiple Outcomes

OK, this is my first posting and is my reason for being here. I apologise up front that this will be a long posting. Sorry. And I’ve probably not best described my question in the title. 

I'm looking at betting/chance on football matches. I keep statistics and while the sample is currently small it has started to show an interesting trend that causes me to ask a maths/probability question. First I will explain the background (you can skip the next 3 paragraphs if you are familiar with football betting) ...

There is a bet that is available from most bookmakers called Double Chance (DC). With a normal match bet book there are three outcomes Home Team Win (H), Draw (D) and Away Team Win (A). With a DC bet you bet that 1 of 2 of the 3 outcomes will result i.e. HD, DA or HA. Because of this the odds are usually quite low as you would expect. To take a current match West Ham v’s Chelsea the DC odds for the game are currently HD=2.45, DA=1.17 and HA=1.29 respectively (I have shown the odds in ‘decimal’ format so for the HD outcome if you staked £1 and the result was either a Home Win or a Draw the bookmaker would return £2.45 including your original stake or £0 (nothing) if Chelsea, the Away Team win the match and similarly for the Draw or Away Win outcomes).

These odds can be converted into a chance by taking the reciprocal (OK, not 100% because there is a bookmaker margin). So the bookmaker’s assessment of the chances of each outcome is 1/2.45, 1/1.17 and 1/1.29 giving 40.81%, 85.47% and 77.52% totalling 203.80% (in a pure situation the total of the chances would be 200% because 2 of the 3 outcomes will result from a match, the 3.80% is the bookmakers margin).

Now because of the relatively low odds (and high chance) many bets combine multiple bets into Doubles, Trebles, Fourfolds and more and this is where my question comes in. The odds for a Double, Treble and Fourfold is the product of the odds for each outcome forming the multiple bet. For example if the outcome chosen for one match is say 1.20 and for the second match is 1.25 the odds for the Double would be 1.50 (1.20 x 1.25) and similarly with 3 outcomes for a Treble and 4 outcomes for a Fourfold.

I have recently monitored matches, odds and outcomes. Since 18th Feb I have monitored 112 matches grouped around the day the match is to be played and if a large number of matches on any day (Saturday being the obvious one) I have created sub-groups using nothing more scientific than kick-off time. I select my matches based on the odds of the favourite outcome of either a HD or DA outcome compared to the other (I ignore the HA outcome). I have attached a zipped Excel file to this thread and would describe the data as follows.

The averages for the groups are:

Average size of a group is 7 (range 4 – 10). Note: (there was one day with only 2 matches but that’s not to be repeated
Minimum odds of a group is 1.16 (range 1.06 – 1.20)
Maximum odds of a group is 1.32 (range 1.25 – 1.39)

If these matches were arranged as Single Match bets, Doubles (2 match combinations from a group), Trebles (3 match combinations from a group) and Fourfolds (4 match combinations from a group) I get the following statistics assuming a level stake of 1 for each bet:

Singles: 112 bets, Profit +1
Doubles: 413 bets, Profit -19
Trebles: 960 bets, Profit +380
Fourfolds: 1524 bets, Profit +407

And to make the comparison more equal and comparable, let’s assume a level stake of £10 for the Single bets and that the total staked for each of the Doubles, Trebles and Fourfolds is the same. The comparison now reads:

Singles: 112 bets @ £10.00 = £1,120.00 Staked … Profit +£7.20
Doubles: 413 bets @ £2.71 = £1,119.23 Staked … Profit -£51.96
Trebles: 960 bets @ £1.17 = £1,123.20 Staked … Profit +£444.05
Fourfolds: 1524 bets @ £0.73 = £1,112.52 Staked … Profit +£296.88

So, my question is this … Is there a plausible, logical, mathematical reason why the minimal profit of Single bets or the actual loss of Double bets should turn into the comparatively spectacular profits of the Treble and Fourfold bets or is it just a fluke of the match outcomes?

I would be interested in any thoughts from those with a much higher knowledge of probability mathematics than my lowly A level teachings of 40 years ago!
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File Type: zip MultipleBets.zip (14.5 KB, 0 views)
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August 7th, 2017, 11:07 PM   #2
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August 12th, 2017, 11:44 PM   #3
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Hi Skipjack and thanks for your interest. I've rather passed on from the original question, it would seem to have been more a statistical coincidence/anomaly (unless the subsequent data is a statistical coincidence! ) but I do have something that has come out of the subject ...

If I have 5 mutually exclusive events and the probability of a favourable outcome from each Event is ... P1, P2, P3, P4 and P5 what would be the formula for the probability of only 1 favourable outcome, 2 favourable outcomes etc. up to all 5 favourable outcomes.

Take 1 favourable outcome, is it ...

P1 x (1-P2) x (1-P3) x (1-P4) x (1-P5)
plus
P2 x (1-P1) x (1-P3) ... etc
plus
P3 ... etc
plus
P4 ... etc
plus
P5 ... etc

In words ...
the P that event 1 is favourable and all other events are not favourable
plus
the P that event 2 is favourable and all other events are not favourable
etc for P3, P4 and P5 being favourable

and then for 2 favourable outcomes it would be ...
P1 x P2 x (1-P3) etc as before

and similarly for 3 and 4 favourable outcomes

finishing with P1 x P2 x P3 x P4 x P5 for all 5 outcomes to be favourable.

I covered basic probability about 50 years ago and not used it since so just looking for confirmation or correction!

Last edited by skipjack; August 13th, 2017 at 12:11 PM.
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