How can I create a matrix for probability formulas so that I can eliminate impossible combinations and thereby increase the total probability factor?
I received a solution here at EE for this question:
Which would the probability formulas be for calculating the difficulty in percentage for 6 different types of wagering?
And this was the answer:
Place: 1/4=25 %
Winner: 1/12=8,3 %
Twins: 1/66=1,5 %
DD: 1/144=0,7 %
Trio: 1/1320
Top 7: 1/3991680
Among these combinations, there are many that are logically and statistically not possible at all, such as that for the wagering type Tvilling in a race with 12 horses, the horse with the 11th lowest odds and the 12th lowest odds would end as the first two horses in the race. So I wonder if, with help from Excel's matrix layout, it would be possible to create formulas in a matrix for each wagering type, and then mark combinations in this matrix as impossible, and Excel then would calculate a new total probability factor.
For example, the total probability factor now for Twins is 1/66 (1.5 %) to pick which two horses will end on the first two places in the race. Let's say I decide, on the basis of statistics and logics, that it's impossible that the horses with 11th and 12th highest odds can win, so I want to mark this combination as impossible in the matrix, and then Excel should re-calculate a new total probability factor. If this is possible to do for each of the different wagering types, I could easily increase the probability factor for picking a winning combination, for example for Twins I might be able to increase the probability factor from 1/66 (1.5 %) to 1/45 (2.2 %), or even more.
Which would the probability formulas be for calculating the difficulty in percentage for 6 different types of wagering?
And this was the answer:
Place: 1/4=25 %
Winner: 1/12=8,3 %
Twins: 1/66=1,5 %
DD: 1/144=0,7 %
Trio: 1/1320
Top 7: 1/3991680
Among these combinations, there are many that are logically and statistically not possible at all, such as that for the wagering type Tvilling in a race with 12 horses, the horse with the 11th lowest odds and the 12th lowest odds would end as the first two horses in the race. So I wonder if, with help from Excel's matrix layout, it would be possible to create formulas in a matrix for each wagering type, and then mark combinations in this matrix as impossible, and Excel then would calculate a new total probability factor.
For example, the total probability factor now for Twins is 1/66 (1.5 %) to pick which two horses will end on the first two places in the race. Let's say I decide, on the basis of statistics and logics, that it's impossible that the horses with 11th and 12th highest odds can win, so I want to mark this combination as impossible in the matrix, and then Excel should re-calculate a new total probability factor. If this is possible to do for each of the different wagering types, I could easily increase the probability factor for picking a winning combination, for example for Twins I might be able to increase the probability factor from 1/66 (1.5 %) to 1/45 (2.2 %), or even more.
ASKER CERTIFIED SOLUTION
membership
This solution is only available to members.
To access this solution, you must be a member of Experts Exchange.
ASKER
I need to re-formulate this, let's take Twins as example:
We assume all horses have equal chance of being 1st or 2nd, no matter the order (or we can still use your first matrix and use those numbers, and if necessary change to so all horses have equal chances and then when needed change for each individual horse so they don't have equal chances), which for a field of 12 horses would give me the probability factor 1/66th (1.5 %) chance to pick the two horses that end 1st and 2nd in the race. I might for example choose horse number 5 and horse number 6, and then the chance/odds/probability for this to happen is 1/66 (1.5 %).
Then, lets assume I decide horse number 11 has no chance at all to end 1st or 2nd, so eliminate this horse from the matrix. Then the probability factor should increase from 1/66th to ? (? %). Then I might decide more horses can be completely eliminated, for example horse number 2 and horse number 4 and horse number 7.
It seems there must be some sort of separation between two probability factors:
1. The probability between the horses, for example that horse 1 has a probability of 0.15 before the other horses to end 1st or 2nd, whereas horse 2 has a probability of 0.07 to end 1st or 2nd, etc. with a total of 100 %.
2. The probability to pick the two horses that make out the Twins, which is 1/66th (1.5 %) before I have eliminated horses in the matrix as impossible to end 1st or 2nd, which should increase this probability factor.
This is the basic relation:
If all horses would have equal chance of getting 1st or 2nd in the race, then each horse would have 1/6th (16.7 %) chance of getting 1st or 2nd in the race.
For me to pick the two horses that end up 1st and 2nd in the race, the probability for me to succeed with this is 1/66th (1.5 %) chance, unless I decide that some horses should be eliminated as I regard them as incapable of ending up 1st or 2nd, in which my probability factor would increase.
The most tricky part is to calculate the probability for Trio and Top 7:
For Trio, I might decide that horse number 5 definitely will end up as 1st or 2nd only, and that horse number 6 can not win the race but end up on all other places (2nd, 3rd).
For Top 7, I might decide that horse number 2 only can end up as 6th or 7th, and that horse number 9 only can end up as 1st or 2nd.
I found some examples of probability matrixes here:
https://www.google.co.uk/search?q=%22probability+matrix%22&tbm=isch&imgil=xm0_biETw3caZM%253A%253BD62B7l1PCtbY2M%253Bhttps%25253A%25252F%25252Fwww.slideshare.net%25252FMulti_Act%25252Fcrystal-ball-for-nifty-using-transition-probability-matrix&source=iu&pf=m&fir=xm0_biETw3caZM%253A%252CD62B7l1PCtbY2M%252C_&usg=__r1GSOwjGjTq1TSSNFbbk22a5tCY%3D&biw=1366&bih=638&ved=0ahUKEwimgMzY9qTVAhWDxLwKHW9uDvgQyjcIUg&ei=sHt3WaboAoOJ8wXv3LnADw#imgrc=xm0_biETw3caZM:
We assume all horses have equal chance of being 1st or 2nd, no matter the order (or we can still use your first matrix and use those numbers, and if necessary change to so all horses have equal chances and then when needed change for each individual horse so they don't have equal chances), which for a field of 12 horses would give me the probability factor 1/66th (1.5 %) chance to pick the two horses that end 1st and 2nd in the race. I might for example choose horse number 5 and horse number 6, and then the chance/odds/probability for this to happen is 1/66 (1.5 %).
Then, lets assume I decide horse number 11 has no chance at all to end 1st or 2nd, so eliminate this horse from the matrix. Then the probability factor should increase from 1/66th to ? (? %). Then I might decide more horses can be completely eliminated, for example horse number 2 and horse number 4 and horse number 7.
It seems there must be some sort of separation between two probability factors:
1. The probability between the horses, for example that horse 1 has a probability of 0.15 before the other horses to end 1st or 2nd, whereas horse 2 has a probability of 0.07 to end 1st or 2nd, etc. with a total of 100 %.
2. The probability to pick the two horses that make out the Twins, which is 1/66th (1.5 %) before I have eliminated horses in the matrix as impossible to end 1st or 2nd, which should increase this probability factor.
This is the basic relation:
If all horses would have equal chance of getting 1st or 2nd in the race, then each horse would have 1/6th (16.7 %) chance of getting 1st or 2nd in the race.
For me to pick the two horses that end up 1st and 2nd in the race, the probability for me to succeed with this is 1/66th (1.5 %) chance, unless I decide that some horses should be eliminated as I regard them as incapable of ending up 1st or 2nd, in which my probability factor would increase.
The most tricky part is to calculate the probability for Trio and Top 7:
For Trio, I might decide that horse number 5 definitely will end up as 1st or 2nd only, and that horse number 6 can not win the race but end up on all other places (2nd, 3rd).
For Top 7, I might decide that horse number 2 only can end up as 6th or 7th, and that horse number 9 only can end up as 1st or 2nd.
I found some examples of probability matrixes here:
https://www.google.co.uk/search?q=%22probability+matrix%22&tbm=isch&imgil=xm0_biETw3caZM%253A%253BD62B7l1PCtbY2M%253Bhttps%25253A%25252F%25252Fwww.slideshare.net%25252FMulti_Act%25252Fcrystal-ball-for-nifty-using-transition-probability-matrix&source=iu&pf=m&fir=xm0_biETw3caZM%253A%252CD62B7l1PCtbY2M%252C_&usg=__r1GSOwjGjTq1TSSNFbbk22a5tCY%3D&biw=1366&bih=638&ved=0ahUKEwimgMzY9qTVAhWDxLwKHW9uDvgQyjcIUg&ei=sHt3WaboAoOJ8wXv3LnADw#imgrc=xm0_biETw3caZM:
SOLUTION
membership
This solution is only available to members.
To access this solution, you must be a member of Experts Exchange.
SOLUTION
membership
This solution is only available to members.
To access this solution, you must be a member of Experts Exchange.
ASKER
Thanks for your comments Alan, I'll read them through closer today and try to clarify those sections I missed out on clarity.
About Trio and Top 7 Pro, I just want to add for later dealing with it (after we've come to a conclusion for the 'on the nose' scenario), that probably I need to evaluate each horse's probability to end up on each individual place, which for Trio is 1-2-3, and for Top 7 is 1-2-3-4-5-6-7.
So for Trio, I need to evaluate probability for horse 1 to end up as 1st, as 2nd, and as 3rd (e.g. probability for horse 1 as 1st is 0.35, probability for horse 1 as 2nd is 0.07, probability for horse 1 as 3rd is 0.05), etc. for horse 2, horse 3 etc. And do the same for Top 7, but for 1-2-3-4-5-6-7.
About Trio and Top 7 Pro, I just want to add for later dealing with it (after we've come to a conclusion for the 'on the nose' scenario), that probably I need to evaluate each horse's probability to end up on each individual place, which for Trio is 1-2-3, and for Top 7 is 1-2-3-4-5-6-7.
So for Trio, I need to evaluate probability for horse 1 to end up as 1st, as 2nd, and as 3rd (e.g. probability for horse 1 as 1st is 0.35, probability for horse 1 as 2nd is 0.07, probability for horse 1 as 3rd is 0.05), etc. for horse 2, horse 3 etc. And do the same for Top 7, but for 1-2-3-4-5-6-7.
SOLUTION
membership
This solution is only available to members.
To access this solution, you must be a member of Experts Exchange.
ASKER
I begin to see your point now with separation of 'overall probability' and 'probability given that'. It seems like you mean the same thing as I mean with my previous comment about this (ID: 42228953):
My 'given that probabilities' consist of my own constants assigned to each horse regarding its chance to win the race (my own odds that I assign, not the racing track's odds). The 'overall probability' consists of the probability that a certain combination should be the turn out of the race, based mathematically on number of possible combinations. But it's primarily in this last case, even though not acceptable from a mathematician's point of view, I want to eliminate horses based on statistics and logics from possible combinations in order to increase the 'overall probability'.
Regarding your comment
this is exactly what I intend to do, to deviate from the true racetrack odds as I set my own odds based on my evaluation of each horse's chance in the race to end up on a certain place. be it 1st, 2nd or 3rd for example.
About 'overall probability' and 'given that probabilities', is there any way to link these together somehow in the same probability matrix?
It seems there must be some sort of separation between two probability factors:
1. The probability between the horses, for example that horse 1 has a probability of 0.15 before the other horses to end 1st or 2nd, whereas horse 2 has a probability of 0.07 to end 1st or 2nd, etc. with a total of 100 %.
2. The probability to pick the two horses that make out the Twins, which is 1/66th (1.5 %) before I have eliminated horses in the matrix as impossible to end 1st or 2nd, which should increase this probability factor.
My 'given that probabilities' consist of my own constants assigned to each horse regarding its chance to win the race (my own odds that I assign, not the racing track's odds). The 'overall probability' consists of the probability that a certain combination should be the turn out of the race, based mathematically on number of possible combinations. But it's primarily in this last case, even though not acceptable from a mathematician's point of view, I want to eliminate horses based on statistics and logics from possible combinations in order to increase the 'overall probability'.
Regarding your comment
Each time you eliminate a horse that has some (albeit small) chance of being placed, if you adjust all the other odds upwards to continue to sum to 100%, you will further distort the true odds.
this is exactly what I intend to do, to deviate from the true racetrack odds as I set my own odds based on my evaluation of each horse's chance in the race to end up on a certain place. be it 1st, 2nd or 3rd for example.
About 'overall probability' and 'given that probabilities', is there any way to link these together somehow in the same probability matrix?
ASKER
There are two main purposes with the probability matrixes:
1. To adjust the 'overall probability' for each of the 6 types of wagering each racing day depending on statistical factors such as that on a certain race track over a certain distance horses with post position 12 almost never end up on 1st or 2nd place. Then the 'overall probability' will increase.
2. In the same race, to find the leverage odds (the official racetrack odds) where a Winner bet corresponds to the same profit as a Twins bet would give (and the same for the other types of wagerings, such as the relation between Top 7 and Trio, Top 7 and Winner, Trio and DD/LD, etc). The odds for a Winner bet is always lower than the odds for a Twins bet because a Winner bet is easier (higher 'overall probability'), so at which odds (the leverage odds) is this difference in 'overall probability' levelled out (at which odds will the difference in probability be compensated for completely)? Isn't the formula 'Overall probability 2'/'Overall probability 1'? I think that when calculating the leverage odds, I also need to take account of the track's share of the total turnover for each type of wagering, which are these:
Place: 20%
Winner: 20%
Twins: 20%
DD: 25%
Trio: 25%
Top 7: 35%
1. To adjust the 'overall probability' for each of the 6 types of wagering each racing day depending on statistical factors such as that on a certain race track over a certain distance horses with post position 12 almost never end up on 1st or 2nd place. Then the 'overall probability' will increase.
2. In the same race, to find the leverage odds (the official racetrack odds) where a Winner bet corresponds to the same profit as a Twins bet would give (and the same for the other types of wagerings, such as the relation between Top 7 and Trio, Top 7 and Winner, Trio and DD/LD, etc). The odds for a Winner bet is always lower than the odds for a Twins bet because a Winner bet is easier (higher 'overall probability'), so at which odds (the leverage odds) is this difference in 'overall probability' levelled out (at which odds will the difference in probability be compensated for completely)? Isn't the formula 'Overall probability 2'/'Overall probability 1'? I think that when calculating the leverage odds, I also need to take account of the track's share of the total turnover for each type of wagering, which are these:
Place: 20%
Winner: 20%
Twins: 20%
DD: 25%
Trio: 25%
Top 7: 35%
SOLUTION
membership
This solution is only available to members.
To access this solution, you must be a member of Experts Exchange.
ASKER
I could see now that I have to make two matrixes; one matrix A for each horse's probability and one matrix B for my own probability to pick the right horses conditioned on matrix A:
First, I use your matrix (matrix A) in your comment ID: 42228849 for setting each horse's probability/odds to be among the first two (in the case of the wagering form Twins).
Then, I must use a new matrix (matrix B) for calculating MY probability to pick the two horses in this 12 horse field who will end up first and second in the race. This probability is conditioned on what I select in matrix A.
If I don't change anything in matrix A, and if all horses in matrix A would have equal values (equal chances of being among the first two in the race), then my probability to pick the first two horses in the race would be 1.5 % (1/66th) in the case of the wagering form Twins. But as soon as I make changes in matrix A, and depending on thereafter what two (or more) horses I select to include in my wagering (system), depending on each individual horse's probability percentage, my probability of picking the correct two first horses in the race either decreases (well, not very probable of course) or, most probable, increases (for example from 1.5 % to 15 % if I select to include horses that I have evaluated have a very good chance of being among the first two in the race).
First, I use your matrix (matrix A) in your comment ID: 42228849 for setting each horse's probability/odds to be among the first two (in the case of the wagering form Twins).
Then, I must use a new matrix (matrix B) for calculating MY probability to pick the two horses in this 12 horse field who will end up first and second in the race. This probability is conditioned on what I select in matrix A.
If I don't change anything in matrix A, and if all horses in matrix A would have equal values (equal chances of being among the first two in the race), then my probability to pick the first two horses in the race would be 1.5 % (1/66th) in the case of the wagering form Twins. But as soon as I make changes in matrix A, and depending on thereafter what two (or more) horses I select to include in my wagering (system), depending on each individual horse's probability percentage, my probability of picking the correct two first horses in the race either decreases (well, not very probable of course) or, most probable, increases (for example from 1.5 % to 15 % if I select to include horses that I have evaluated have a very good chance of being among the first two in the race).
EE asked for assistance in closing this question.
ASKER
1. How are the other odds "forced" up to a total of 100 %? I think it's the formula in each cell, isn't it?: =A1/SUM($A$1:$C$12). Probabilitywise so to speak, is this acceptable from a mathematician's view? It's a proper representation of the probability situation, isn't it? I mean, I decide there are certain combinations that are not possible, so then omit these and then calculate a new probability based on this.
2. I want to have equal values in each cell, no problems with that? But now I don't get the pieces together: If, in the case of Twins, I select horse number 5 and horse number 6 as the pair and wager on this, then I would have a 1/66th (1.5 %) chance of winning. How can I spread this figure (1/66th; 1.5 %) equally in each of the 12 cells in the matrix?