Numb3rs 306: Longshot

In this episode the team investigates the death of a man (Danny) at a horce racing track. Danny had been able to pick the winning horse in the last 30 races he had bet on. Had there not been any foul play involved, this would be extremely unlikely (there are typically 10 horses in each race). Danny had been using a system of betting that his girlfriend (who had a Masters degree in Mathematics from Stanford) had developed to pick out the second most likely horse to win a race. She developed this system in order to take advantage of the higher expectation of betting on the second best horse arising from parimutual betting. When Danny noticed that it was the second best horse that had been winning consistantly, he suspected that the races were being fixed, and took advantage of it.

### Parimutual Betting

Parimutual betting is performed at horse tracks all over the globe. Rather than a professional gambler determining the proper odds for a race (based on knowledge of the horses and experience), this system allows bettors to determine the odds by themselves. We’ll give a simple example of a payout system where only bets for the winning horse pay off money. Suppose that the track is running a 5 horse race, and say that bets are placed as follows.

Horse 1 $4000 | Horse 2 $1000 | Horse 3 $9000 | Horse 4 $2000 | Horse 5 $14000 |

The total amount bet is $30,000. First the house takes its cut (to be concrete we’ll assume this is 15%, leaving $27,000). Then anyone who bet $1 on horse 3 will get $27000/9000 = $3. We can see that Parimutual betting guarantees the house a profit, since it always pays out exactly what it took in minus the cut (of course, we’re assuming the cut covers the cost of operation and no laws interfere). Now, why is it that Danny and his girlfriend were concerned with finding a way to pick the second most likely to win? Since the odds are set by the bettors they may not actually reflect the true odds of a horse winning a race. Most bettors will concentrate on the best horse and place there bets there, often forgetting about the second best. Suppose that this had happened in our race, and that the actual odds for each horse to win had been

Horse 1 13% | Horse 2 4% | Horse 3 35% | Horse 4 8% | Horse 5 40% |

Then our expectation of a $1 bet on Horse 5 is 27000/14000*.4 – 1*.6= $.171 and the expected value of a bet on Horse 3 is $27000/9000*.35-.65*1 = $.4, which is far greater than that of Horse 5. (See here or here for a reminder of what an expected value is.) Now, when the track owners were fixing the races, they would pick the second best horse to be the winner because it would cause less suspiscion than making a worse horse win.

### Arbitrage

Simply put, arbitrage is risk-free profit. That is, when we invest some amount of money, we will lose money with probability 0 and have some positive chance of gaining money. A market that is free of arbitrage opportunities is called an *Efficient Market*, and markets with arbitrage opportunities are called *Inefficient Market*. In *Longshot* Charlie claims (correctly) that the horse race gambling market is inefficient. Example, suppose for simplicity that there are only 2 horses in some race and that two bookmakers disagree on the odds of the outcomes, say

Now if we place a $1000 bet on Outcome 1 with Bookmarker 2 and a bet of $1000*1.5/3.56=$421.35 on Outcome 2 with Bookmaker 1, we will have taken up an arbitrage strategy. Let’s make sure. If Outcome 1 happens we win $1000*1.5 from Bookmaker 2, if Outcome 2 happens we win $$421.35*3.56=$1500. Since we invested $1421.35 we are guaranteed a profit of $88.65 either way.

Every wonder why the price of a stock on the NYSE and its corresponding futures contrace on the Chicago Mercantile Exchange are the same? The answer is arbitrage. If there were ever a difference in the price of some particular stock, one could buy that stock in the cheaper marker and sell it at the same time in the more expensive market. Many investment companies have complicated computer programs that track opportunities for arbitrage (many are more complicated than this simple example) and try to take advantage of them as quickly as possible. For more info on arbitrage see here.

Numb3rs 306: Longshot In this episode the team investigates the death of a man (Danny) at a horce racing track. Danny had been able to pick the winning horse in the last 30 races he had bet on.*Horse Racing Stats*: The Best, Most Common Winning Horse Numbers, Triple Crown History

*Horse Racing Stats*: The Best, Most Common Winning Horse Numbers, Triple Crown History

### By Ion Saliu, *Founder of Horse Betting Mathematics*

Published April 27, 2012.

Reports created by **FrequencyRank** , horse racing statistical software.

The statistical reports rank the horse numbers by their frequency . The numbers are listed in __descending order__, from the most frequent (the *hottest* or *best numbers*) to the least frequent numbers (the *coldest* or the *worst numbers*). The reports show the number of hits ; i.e. amount of races in which that particular horse number was a winner.

The ranking by frequency is performed two ways:

1) Regardless of position;

2) Position by position.

I’ve been looking for a long time to find all the results of the Triple Crown races. However, I did not want the results expressed as *horse names*, but *horse numbers* as in *program numbers*. Sometimes, the *program numbers* coincide with the *post positions*. The bettors bet on *program numbers*, not names. For example: The winning trifecta at the 2011 Kentucky Derby was *16 – 19 – 13*.

It is extremely difficult to find such data. After serious Internet search, I had some success. I have now the entire history of the Kentucky Derby. I have 109 *by-horse-numbers* races at the Belmont Stakes. They had a couple of years without running. Also, in the 19th century, data for Belmont Stakes only shows the results by horse names.

I haven’t been nearly as successful with the Preakness Stakes. That race is, indeed, the poor relative of the two big Triple Crown races! The official Preakness site does not keep records like Churchill Downs (Kentucky Derby) and Belmont Park (Belmont Stakes). Here are PDF samples from 2011 Kentucky Derby and Belmont Stakes: the winners by horse number (program), showing also the post positions (PP):

Despite my efforts, I only have 24 *by-horse-numbers* races at the Preakness Stakes (2011 – 1988). Before that (1988), the history is available only by horse names. I will publish the winning trifectas between 2011 – 1988 on a separate page, like for Kentucky Derby and Belmont States.

Even if the Kentucky Derby and the Belmont Stakes have large databases, the results are badly biased. Many, many races in the past had very few horses. I saw many races with just 3 horses. A couple of races only had 2 horses! The results are more meaningful in the modern era. The Kentucky Derby, for example, tried very hard to assure 20 horses per race beginning 1990s.

I created statistical reports for three periods (number of races):

1) the entire available history;

2) beginning with the legendary horse *Secretariat* (1973 to 2011, 38 races);

3) beginning with 1990 (21 races to 2011).

A few facts for the Kentucky Derby –

1.1. The entire history (136 races) — *regardless of the 3 positions*

#1 = 53 times; #3 = 50 times; #2 = 40 times (extreme bias because of the past);

1.2. The entire history (136 races) — *position by position*

1st position: #1 = 23 times;

2nd position: #5 = 23 times;

3rd position: #2 = 22 times.

2.1. The last 38 races — *regardless of the 3 positions*

#2 = 12 times; #3 = 11 times; #8 = 10 times;

2.2. The last 38 races — *position by position*

1st position: #4 = 7 times;

2nd position: #5 = 7 times;

3rd position: #2 = 7 times.

3.1. The last 21 races — *regardless of the 3 positions*

#5 = 7 times; #2 = 6 times; #8 = 6 times;

3.2. The last 21 races — *position by position*

1st position: #6 = 4 times;

2nd position: #5 = 5 times;

3rd position: #2 = 4 times.

Looks like #2 and #5 fared the best in all three periods of time at the Kentucky Derby. A good pool of numbers based on the last two periods of time: *2 – 4 – 5 – 6*.

A few facts for the Belmont Stakes –

1.1. The entire available history (109 races) — *regardless of the 3 positions*

#1 = 58 times; #2 = 47 times; #3 = 47 times (extreme bias because of the past);

1.2. The entire available history (109 races) — *position by position*

1st position: #1 = 25 times;

2nd position: #3 = 20 times;

3rd position: #2 = 19 times.

2.1. The last 38 races — *regardless of the 3 positions*

#2 = 16 times; #5 = 14 times; #3 = 13 times;

2.2. The last 38 races — *position by position*

1st position: #5 = 6 times;

2nd position: #2 = 8 times;

3rd position: #4 = 5 times.

3.1. The last 21 races — *regardless of the 3 positions*

#2 = 9 times; #4 = 8 times; #5 = 7 times;

3.2. The last 21 races — *position by position*

1st position: #2 = 3 times;

2nd position: #8 = 4 times;

3rd position: #5 = 4 times.

Looks like #2 fared the best in all three periods of time at the Belmont Stakes. A good pool of numbers based on the last two periods of time: *2 – 4 – 5 – 8*.

Looks like the best horse numbers based on the last two periods of time, at both Kentucky Derby and Belmont Stakes, are *2 – 4 – 5*. Let’s not jump to the conclusion that the horse races are rigged!

The results (in numerical format) and the statistical analyses for the Kentucky Derby, Preakness Stakes, and the Belmont Stakes are available on three Web pages. I may or may not update those resources. You need to keep your own records from now on. Just add the winning trifecta after the respective race ends.

The statistical reports and result files (winning trifectas) will be updated soon after the respective Triple Crown race ends. Such resources are not available for the Preakness Stakes at this time.

• BrightH3 / Ultimate Horse Software are the most powerful software packages for betting at the horse tracks, especially OTW (off-track wagering). The software performs the most sophisticated statistical analyses and it generates straight trifectas with the highest chance to win any race.

Horse-racing software ranks all horse numbers by frequency from best to worst statistically, showing the most common winning post positions. ]]>