The Maths of Multiples, Part 2

In my previous article, the discussion moved to each-way doubles and how the makeup of certain races / markets has the potential to give punters an edge over bookmakers, certainly in terms of the place part of such a bet, writes Dave Renham. You can read that article here. In this follow up, we are going to dive deeper into the world of the ‘each-way double’.

For these bets to be profitable in the long run, we ideally need to have achieved a value price on both of the two selections, in terms of the win part as well as the place part, for every bet that has been struck. In other words, the actual chance for both parts of the bet for both selections are higher than the percentage chances on offer from the bookmaker. Technically we could also make a long-term profit if one of the two parts of the bet always offers enough value to compensate for the part that does not. It goes without saying that achieving any sort of edge is far from easy.

To try and illustrate this in numbers let us imagine an each-way double where we have place terms of one quarter the odds, and the price on both selections is 5.0 (4/1). The true percentage odds for a horse priced 5.0 to win is 20%; the true percentage chance for a place with quarter the odds stands at 50%. (N.B. a ‘win’ constitutes a ‘place’, so effectively ‘place’ means ‘win or place’). Therefore, if both horses are priced over 5.0 to win, we have a value bet.

To help crunch the numbers, I have created an excel spreadsheet to simulate a series of each-way doubles with a specific price / percentage chance for each selection. The number of simulated each-way selections for each series of bets has been set to 1000. This is a huge number of races but due to each-way doubles rarely achieving two wins, it makes sense to use such a large number. Ultimately it is easy to tweak the number of bets/races for each simulated series, so I could adjust the number of races up or down. Within each simulation I can adjust the win/place percentage chance of these ‘runners’ (above and below the ‘true’ percentage chance) to examine the long-term outcome from a profit/loss/returns perspective.

So, to kick off I'll return to the example of the two 5.0 priced runners given in the third paragraph, where we have place terms of one quarter the odds. If we assume each 5.0 price is always the ‘true’ price, then over 1000 bets we would break even. The maths look like this based on a £1 each-way stake (£2 in total) on these 1000 each-way doubles:

Races with two wins – 40
Races with ‘place’ wins (e.g. Win/Place or Place/Win or Place/Place) – 210
Races where the bet was a loser (e.g. at least one horse unplaced) - 750

Thus...

The 40 successful win bets would have paid £960 profit and £120 profit on the place part of the bet;
The 210 winning place bets would have paid £630 profit for the place part but would have lost the £210 placed on the win part of the bet, leaving a profit of £420.
The 750 losing bets would have seen a loss of both the win stake and the place stakes equating to £1500.

Adding the profits of £960, £120 and £420 we end up in credit to the tune of £1500, but subtracting the losing bet total of £1500 we arrive at that break-even situation as mentioned above.

This 5.0 / 5.0 example shows what happens when the horses win and place as often as they theoretically should do based on their odds / percentage chance. In this case we know already that a true 5.0 shot should win 20% of their races and win or be placed in 50% of them. So, what happens when these percentage chances differ from the ‘true’ percentage chances? This is where my spreadsheet comes in to save time.

The first graph will look at the effect that different win and placed percentages have on the long-term Return on Investment figure (ROI%). Remember we will be theoretically placing 1000 each-way doubles on two horses priced 5.0 with quarter the odds a place. In order to help explain the graph, the table below shows how I have adjusted the win and place percentages:

There is a neat symmetry to these results. As is shown, if we could increase the average win chance by 2% (from 20% to 22%) and the placed percentage also by 2 (from 50% to 52%) the long-term return would be a pleasing 14% or 14p in the £.

Let’s look at another example. This time we will assume both horses are priced 6.0 (5/1) but the place terms are 1/5 of the odds. I have tweaked/changed the percentages in exactly the same way as I did in the first example. For the record the true win percentage chance here is 16.7% (to 1 decimal place), with the place chance being 50%. Here is what I found:

We get a similar pattern as one would expect although not quite the exact symmetry of the previous example. It should be noted at this juncture that the 1000 bet simulation using this particular price point could show that a horse has won 39.89 times in the 1000 races! Clearly this is not possible, so I have simply rounded the ‘expected’ win or placed results to the nearest whole number. Therefore, one could argue that the 7.8% ROI for let’s say the W17.7 P51 group is not 100% accurate, but it is as near as doesn't matter in the context of these simulations.

Time now to look at two horses with different prices; I am going to consider horses priced 3.0 (2/1) and 5.0 (4/1), using one quarter the odds a place. This is more difficult to show on one graph because we now have effectively two win percentages and two place percentages being used for each ROI% calculation. Hence, I am going to share two separate graphs.

In the first of the two graphs the ROI%s shown are based solely on the win percentages for each horse changing compared to their true win percentage chances. In this case I have assumed that the place percentage chance has remained the same for both horses for this simulation. The bottom of the graph has a key which I will now explain:

TW stands for the true win percentage chance for each horse based on their prices of 3.0 and 5.0 (which 33.3% and 20%)

TW+1 stands for the true percentage win chance +1% (34.3% and 21%)

TW+2 stands for the true percentage win chance +2% (35.3% and 22%)

TW+3 stands for the true percentage win chance +3% (36.3% and 23%)

TW-1 stands for the true percentage win chance -1% (32.3% and 19%)

TW-2 stands for the true percentage win chance -2% (31.3% and 18%)

TW-3 stands for the true percentage win chance -3% (30.3% and 17%)

To be clear, the graph below shows the effect on the ROI% of only theoretical changes in the long-term win percentages. All place calculations are based on their true percentage place chances which for the 3.0 priced horse is 66.7% (to 1 decimal place) and for the 5.0 priced horse it is 25%.

Assuming we can improve our percentage chance of each horse winning by 3% compared to their true chance we would make close to 13p in the £ over the long term. Conversely, if the percentages for both horses dropped by 3% we would be losing just over 11p in the £.

The second graph will show theoretical changes in the place percentage while keeping the win percentages at their true figures. I will again use +1, +2 idea but this time I will use TP meaning true place percentage:

As we can see a change of 1%, 2% or 3% in the placed percentages has less influence on the ROI%. However, if we can gain a place edge while keeping parity in terms of the win percentage then we will make money over the longer term. Clearly if we can gain both a win edge and place edge your profits should mount up nicely.

It is time to look at some ‘real life’ data now, based on actual horse racing results and prices going back to 2010. I looked at all 12-runner handicaps over this time frame which covered over 90,000 runners. I looked at the prices of different runners and extrapolated their actual long-term win and placed percentages. I then used these figures to work out the potential returns on a series of 1000 each-way doubles with said percentage chances. As I was simulating 12-runner handicaps for each of the 1000 races, the place terms were set at one quarter the odds. Here is what I found:

This does not make for pleasant reading. The reason behind such poor figures is, of course, the bookmaker edge I discussed in the first article. The average prices of the runners do not reflect their true chance of winning, or indeed placing.

Readers may note that the 5.0 / 5.0 figures see a loss of 16.6%. If we go back to the first graph in this article which examined a 5.0 price / 5.0 price simulation, we can see that when the win% was set at 18% and the place at 48%, this lost 14%. Given the real-life figures gave us an ROI% just below that at 16.6% it is no surprise to know that the actual win and placed percentages based on the 2010-2024 average figures were just below these at 17.97% for the win and 46.67% for the place.

Understanding how probability works is important when it comes to betting singles, but I would argue it is even more important when it comes to combining two or more horses in a bet. Probability can be a complex subject at a higher level but, fortunately, for betting purposes it is relatively straightforward. Just remember, for those of us betting doubles, trebles, fourfolds, and so on, we need to multiply the percentage chances together to give the overall percentage chance.

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I hope this has been an enjoyable and informative piece, and that it has tied in neatly with the previous one. Each-way doubles are bets that are worth considering given the right conditions of price to chance and race shape. I am in two minds about whether to dig a bit deeper with the potential to research and write a third piece. I will play around with some ideas in the next few weeks so watch this space!

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Before I go, if anyone is interested in a more complex probability challenge please read on.

I would like to share a famous problem that I used to pose to my older students when I taught in schools. It is a version of what is known as the ‘Birthday Paradox’.

Imagine 30 children in a room all aged 12. What is the probability that at least two of them will share the same birthday? 

Now for those who have not heard this paradox before, then I urge you to take an educated guess at the chance of this occurring before reading on.

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When I posed this question to my classes over the years, their thinking generally went along the lines that there are 30 children, there are 365 days in a year and hence they divided 30 into 365. Having said that, most of them ‘rounded’ 365 to 300 to make the calculation easier, where 30 divided by 300 gives an answer of 10%. So, I would say 95% of all answers I received were in the ballpark of 10%. However, 10% is way off.

The actual answer is 69.68%.

Be honest now – how many saw that coming?

I am sure there will be a high proportion of readers that will be incredulous that this is the answer. However, if you Google it, you will see that the information above is accurate, and how the answer is calculated.

Until next time...

- DR

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• The ‘Super Six’ NH Jockeys: What Happened Next?
• A Look at Favourites in All-Weather Races
• Kempton 7f Handicaps: Deep Dive