Database question: Average odds by margin of victory

[lansdale]

For anyone who might maintain this kind of stat, what are the average odds of winners by margin of victory? Any kind of further breakdown of this into subsets would, of course, also be great. Thanks for any help.

Cheers,

lansdale

[TrifectaMike]

For anyone who might maintain this kind of stat, what are the average odds of winners by margin of victory? Any kind of further breakdown of this into subsets would, of course, also be great. Thanks for any help.

Cheers,

lansdale

lansadle,

This type of data might be more meaningful, if the odds and margin of victory are considered as two random variables and the raw data presented in a two way relative frequency table.

For example:

Margin of Victory brackets vs Odds Brackets and the number of winners in each joint bracket (.1 - .3 lengths, $.10 -$.50, number of winners, etc). In this way one can analyze joint, marginal and conditional probabilities.

Averages don't reveal very much and neither do marginal probabilities in many instances.

Mike (Dr Beav)

[lansdale]

Hi TM,

Thanks for your reply and apologies re lateness of mine. I take your point (I think) about the best way of looking at this problem. But someone as math-challenged as myself could use some added guidance. You mention bracketing random samples in each of these categories. Would ten of each be an acceptable minimum, as it usually seems to be in your examples? And would it not be clearer to use Venn diagrams to illustrate the nature of the relationships?

Thanks again.

Cheers,

lansdale

[TrifectaMike]

Hi TM,

Thanks for your reply and apologies re lateness of mine. I take your point (I think) about the best way of looking at this problem. But someone as math-challenged as myself could use some added guidance. You mention bracketing random samples in each of these categories. Would ten of each be an acceptable minimum, as it usually seems to be in your examples? And would it not be clearer to use Venn diagrams to illustrate the nature of the relationships?

Thanks again.

Cheers,

lansdale

Hi lansdale,

Venn diagrams are a good visual aid, but a clumsly way to display "real" data.

Let's look at how we would setup a frequency distribution table for the the question you've asked.

We'll define some brackets:

Margin of Victory (lengths)
[0 - 3/4] [1- 1-3/4] [0 - 3/4] [2 - 2-3/4] [3 - 3-3/4] [4 - 4-3/4] [5- 5-3/4] [>5-3/4]

Odds
[.1 - .5] [.6 - 1] [1.1 - 2] [2.1 - 3] [3.1 - 4] [4.1 - 5] [5.1 - 6] [6.1 - 7] [>7]
Depending on the resolution one is seeking and the data permitting, you can have numerous brackets.

Now, we can setup a frequency table ( for the purpose of simplicity, we will keep the table data as counts).

Let's say we have a 2000 races. That is 2000 winners. We can proceed to fill the table.

Note: The data has no reference to reality.

(If someone can make this table align, please do so. Thanks)

[0 - 3/4] [1- 1-3/4] [2 - 2-3/4] [3 - 3-3/4] [4 - 4-3/4] [5 - 5-3/4] [>5-3/4] Total

[.1 - .5] 10 20 30 40 50 70 120 340
[.6 - 1] 10 15 20 30 35 40 64 214
[1.1 - 2] 12 20 20 25 30 35 55 197
[2. 1- 3] 18 25 40 45 45 50 50 273
[3.1 - 4] 20 22 45 45 50 55 60 297
[4.1 - 5] 25 30 50 60 65 60 50 340
[5.1 - 6] 30 25 20 15 15 10 9 124
[6.1 - 7] 40 20 10 10 7 7 6 100
[>7] 50 40 6 6 5 3 3 115
Total 215 217 241 276 302 332 417 2000

The table is read, for example, as follows:

There were 10 winners for the joint brackets of [0 - 3/4] (margin of victory and [.1 - .5] (Odds).

There were 45 winners for the joint brackets of [3 - 3-3/4] (margin of victory)and [2. 1- 3] (Odds).

There were 7 winners for the joint brackets of [5 - 5-3/4] (margin of victory)and [6.1 - 7] (Odds).

What type of questions can this frequency table help us answer about the data?

Here are some questions and answers:

What is the probability of a horse that wins the race has odds, [3.1 - 4] and has a margin of victory, [4 - 4-3/4]?
This is just a joint probability. The number of "odds, [3.1 - 4] and margin of victory, [4 - 4-3/4], divided by the Total = 50/2000 = .03 or 3%

What is the probability of a randomly selected winner has odds, [2. 1- 3]?
This is the total for [2. 1- 3] divided by total races = 273/2000 = .14 or 14%

What is the probability of a randomly selected winner has a margin of victor [5 - 5-3/4]?
This is the total of [5 - 5-3/4] divided by total races = 332/2000 = .17 or 17%.

What is the probability of a randomly selected horse with odds, [1.1 - 2] has margin of victory, [>5-3/4]?
There are a total 55 horses with margin of victory, [>5-3/4] out of 197 horses with odds, [1.1 - 2]. This is 55/197 = .28 or 28%.

Given that a horse has odds, [6.1 - 7], what is the probability it has a margin of victory, [4 - 4-3/4]?
This time you divide 7/302 = .02 or 2%.

As you can see, there are many questions that can be answered by the use of a frequency table.

Mike (Dr Beav)

[lansdale]

Hi lansdale,

Venn diagrams are a good visual aid, but a clumsly way to display "real" data.

Let's look at how we would setup a frequency distribution table for the the question you've asked.

We'll define some brackets:

Margin of Victory (lengths)
[0 - 3/4] [1- 1-3/4] [0 - 3/4] [2 - 2-3/4] [3 - 3-3/4] [4 - 4-3/4] [5- 5-3/4] [>5-3/4]

Odds
[.1 - .5] [.6 - 1] [1.1 - 2] [2.1 - 3] [3.1 - 4] [4.1 - 5] [5.1 - 6] [6.1 - 7] [>7]
Depending on the resolution one is seeking and the data permitting, you can have numerous brackets.

Now, we can setup a frequency table ( for the purpose of simplicity, we will keep the table data as counts).

Let's say we have a 2000 races. That is 2000 winners. We can proceed to fill the table.

Note: The data has no reference to reality.

(If someone can make this table align, please do so. Thanks)

[0 - 3/4] [1- 1-3/4] [2 - 2-3/4] [3 - 3-3/4] [4 - 4-3/4] [5 - 5-3/4] [>5-3/4] Total

[.1 - .5] 10 20 30 40 50 70 120 340
[.6 - 1] 10 15 20 30 35 40 64 214
[1.1 - 2] 12 20 20 25 30 35 55 197
[2. 1- 3] 18 25 40 45 45 50 50 273
[3.1 - 4] 20 22 45 45 50 55 60 297
[4.1 - 5] 25 30 50 60 65 60 50 340
[5.1 - 6] 30 25 20 15 15 10 9 124
[6.1 - 7] 40 20 10 10 7 7 6 100
[>7] 50 40 6 6 5 3 3 115
Total 215 217 241 276 302 332 417 2000

The table is read, for example, as follows:

There were 10 winners for the joint brackets of [0 - 3/4] (margin of victory and [.1 - .5] (Odds).

There were 45 winners for the joint brackets of [3 - 3-3/4] (margin of victory)and [2. 1- 3] (Odds).

There were 7 winners for the joint brackets of [5 - 5-3/4] (margin of victory)and [6.1 - 7] (Odds).

What type of questions can this frequency table help us answer about the data?

Here are some questions and answers:

What is the probability of a horse that wins the race has odds, [3.1 - 4] and has a margin of victory, [4 - 4-3/4]?
This is just a joint probability. The number of "odds, [3.1 - 4] and margin of victory, [4 - 4-3/4], divided by the Total = 50/2000 = .03 or 3%

What is the probability of a randomly selected winner has odds, [2. 1- 3]?
This is the total for [2. 1- 3] divided by total races = 273/2000 = .14 or 14%

What is the probability of a randomly selected winner has a margin of victor [5 - 5-3/4]?
This is the total of [5 - 5-3/4] divided by total races = 332/2000 = .17 or 17%.

What is the probability of a randomly selected horse with odds, [1.1 - 2] has margin of victory, [>5-3/4]?
There are a total 55 horses with margin of victory, [>5-3/4] out of 197 horses with odds, [1.1 - 2]. This is 55/197 = .28 or 28%.

Given that a horse has odds, [6.1 - 7], what is the probability it has a margin of victory, [4 - 4-3/4]?
This time you divide 7/302 = .02 or 2%.

As you can see, there are many questions that can be answered by the use of a frequency table.

Mike (Dr Beav)

Hi TM.

Again much thanks for taking on this problem. The chart is fascinating alone in suggesting many fewer close finsihes that I had thought. In all, the distribution seems consistent with power laws you've mentioned. I will definitely be using this.

One request: after reading so many of your posts on Bayes, and on poker sites, as well, I'd really like to learn more about it. Although there is no 'Bayes for Dummies', could you recommend an intro on this subject for the layman?

Thanks again.

Cheers,

lansdale

[zerosky]

I found this quite useful...


http://www.rigb.org/christmaslectures08/html/activities/learning-from-probabilities.pdf

[SchagFactorToWin]

Hi TM.

Again much thanks for taking on this problem. The chart is fascinating alone in suggesting many fewer close finsihes that I had thought. In all, the distribution seems consistent with power laws you've mentioned. I will definitely be using this.



I believe the chart data is made-up for demo purposes, not from actual data.

[TrifectaMike]

Hi TM.

Again much thanks for taking on this problem. The chart is fascinating alone in suggesting many fewer close finsihes that I had thought. In all, the distribution seems consistent with power laws you've mentioned. I will definitely be using this.

One request: after reading so many of your posts on Bayes, and on poker sites, as well, I'd really like to learn more about it. Although there is no 'Bayes for Dummies', could you recommend an intro on this subject for the layman?

Thanks again.

Cheers,

lansdale

Hi lansdale,

Glad to hear you have found the two way frequency table helpful. I'm more impressed with the fact that you connected the data to the power law distributions over two years ago (some people do read my posts).

I am going to suggest a book, which most people would think is crazy as a subsititute to "Bayesian for Dummies". The book is

"Probability Theory: The Logic of Science" by E. T. Jaynes

This book is a graduate level book. The reason I recommend this book is not for the practical application of Bayes Theorem, but for the purpose of exposing you to Bayes thinking. I recommend reading this book and skip over the math you don't understand. Read it for the purpose of thinking Bayesian.
To truly understand apply Bayesian probability one must become a Bayesian thinker.

"The Theory That Would Not Die: How Baye's Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversey" by Sharon Bertsch McGrayne"

This book will inspire you to take Bayesian Probability seriously and an interesting read. This book will not give you much meat, but it is entertaining and elightening from a historical point of view.

"Basic Probability Theory" by Robert B. Ash

This book covers the basics very nicely. I would say it is a step beyond first year basic probability.

A suggestion: I would concentrate on the discrete form of Bayes Theorem. It is much easier to understand and implement. Instead of probability vectors being a point in n-dimensional space, it is instead m discrete values... and integrals are replaced with summations. Furthermore , instead of utilizing probability density functions as is used if the random variable was continuous, the distribution is now a probability distribution or just a probability.

Good luck,
Mike (Dr Beav)

[lansdale]

Hi lansdale,

Glad to hear you have found the two way frequency table helpful. I'm more impressed with the fact that you connected the data to the power law distributions over two years ago (some people do read my posts).

I am going to suggest a book, which most people would think is crazy as a subsititute to "Bayesian for Dummies". The book is

"Probability Theory: The Logic of Science" by E. T. Jaynes

This book is a graduate level book. The reason I recommend this book is not for the practical application of Bayes Theorem, but for the purpose of exposing you to Bayes thinking. I recommend reading this book and skip over the math you don't understand. Read it for the purpose of thinking Bayesian.
To truly understand apply Bayesian probability one must become a Bayesian thinker.

"The Theory That Would Not Die: How Baye's Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversey" by Sharon Bertsch McGrayne"

This book will inspire you to take Bayesian Probability seriously and an interesting read. This book will not give you much meat, but it is entertaining and elightening from a historical point of view.

"Basic Probability Theory" by Robert B. Ash

This book covers the basics very nicely. I would say it is a step beyond first year basic probability.

A suggestion: I would concentrate on the discrete form of Bayes Theorem. It is much easier to understand and implement. Instead of probability vectors being a point in n-dimensional space, it is instead m discrete values... and integrals are replaced with summations. Furthermore , instead of utilizing probability density functions as is used if the random variable was continuous, the distribution is now a probability distribution or just a probability.

Good luck,
Mike (Dr Beav)

Hi TM,

Much thanks for these recommendations. I've seen the Jaynes book on the shelf of a physicist friend - obviously a landmark. And, from the way it sounds, probably much more accessible for someone like myself, with more of a background in philosophy than math. I saw the McGrayne book reviewed in the Times a couple of months ago, and have intended to check it out. For anyone interested, this is a link to the review-
http://www.nytimes.com/2011/08/07/books/review/the-theory-that-would-not-die-by-sharon-bertsch-mcgrayne-book-review.html?_r=1&scp=1&sq=McGrayne&st=cse

One of the main points made to me about Bayes, by a friend knowledgeable in behavioral economics, is that, although it's regarded as too 'subjective' by its frequentist critics, it in fact more closely mirrors the limits of the human mind in grasping natural phenomena (or 'reality' generally), and that the various frequentist theories and methods offer a degree of precision, which is, in fact, an illusion.

Again, much thanks.

Cheers,

lansdale

[lansdale]

I believe the chart data is made-up for demo purposes, not from actual data.

Hi Schag,

Thanks for your input. As I said, I'm not a math guy, but I understand that much re Bayes. That's why I said 'suggests'.

Cheers,

lansdale

[lansdale]

I found this quite useful...


http://www.rigb.org/christmaslectures08/html/activities/learning-from-probabilities.pdf

Hi Zerosky,

Thanks for this helpful link.

Cheers,

lansdale