Guaranteed Winning Strategy on Horse Betting Odds
Suppose four horses - $A, B, C$, and $D$ - are entered in a race and the odds on them, respectively, are $5$ to $1$, $4$ to $1$, $3$ to $1$, and $2$ to $1.$ If you bet $$1$ on $A$, then you receive $$6$ if $A$ wins, or you realize a net gain of $$5$. You lose your dollar if $A$ loses. How should you bet your money to guarantee that you can always make money no matter which horse wins?
I found this question from Horse Betting Odds - But Guaranteed Win! and made some revision.
I think if I bet my money on A, B, C, and D under the following scenario, such as A:B:C:D = 20:15:12:10. I can always make money. These numbers come from the ratios 1/3 : 1/4 : 1/5 : 1/6 multiplied by 60， where 60 is the lowest common multiple for 3,4,5, and 6.
However, I wonder if there is a formal mathematical way to derive this solution. Is this question related to solving the linear inequality system or the probability?
probability algebra-precalculus systems-of-equations linear-programming
edited Jan 4 at 11:34
user1551
71.8k566125
asked Jan 4 at 9:07
Bratt SwanBratt Swan
1234
add a comment |
Suppose four horses - $A, B, C$, and $D$ - are entered in a race and the odds on them, respectively, are $5$ to $1$, $4$ to $1$, $3$ to $1$, and $2$ to $1.$ If you bet $$1$ on $A$, then you receive $$6$ if $A$ wins, or you realize a net gain of $$5$. You lose your dollar if $A$ loses. How should you bet your money to guarantee that you can always make money no matter which horse wins?
I found this question from Horse Betting Odds - But Guaranteed Win! and made some revision.
I think if I bet my money on A, B, C, and D under the following scenario, such as A:B:C:D = 20:15:12:10. I can always make money. These numbers come from the ratios 1/3 : 1/4 : 1/5 : 1/6 multiplied by 60， where 60 is the lowest common multiple for 3,4,5, and 6.
However, I wonder if there is a formal mathematical way to derive this solution. Is this question related to solving the linear inequality system or the probability?
probability algebra-precalculus systems-of-equations linear-programming
edited Jan 4 at 11:34
user1551
71.8k566125
asked Jan 4 at 9:07
Bratt SwanBratt Swan
1234
add a comment |
Suppose four horses - $A, B, C$, and $D$ - are entered in a race and the odds on them, respectively, are $5$ to $1$, $4$ to $1$, $3$ to $1$, and $2$ to $1.$ If you bet $$1$ on $A$, then you receive $$6$ if $A$ wins, or you realize a net gain of $$5$. You lose your dollar if $A$ loses. How should you bet your money to guarantee that you can always make money no matter which horse wins?
I found this question from Horse Betting Odds - But Guaranteed Win! and made some revision.
I think if I bet my money on A, B, C, and D under the following scenario, such as A:B:C:D = 20:15:12:10. I can always make money. These numbers come from the ratios 1/3 : 1/4 : 1/5 : 1/6 multiplied by 60， where 60 is the lowest common multiple for 3,4,5, and 6.
However, I wonder if there is a formal mathematical way to derive this solution. Is this question related to solving the linear inequality system or the probability?
probability algebra-precalculus systems-of-equations linear-programming
edited Jan 4 at 11:34
user1551
71.8k566125
asked Jan 4 at 9:07
Bratt SwanBratt Swan
1234
Suppose four horses - $A, B, C$, and $D$ - are entered in a race and the odds on them, respectively, are $5$ to $1$, $4$ to $1$, $3$ to $1$, and $2$ to $1.$ If you bet $$1$ on $A$, then you receive $$6$ if $A$ wins, or you realize a net gain of $$5$. You lose your dollar if $A$ loses. How should you bet your money to guarantee that you can always make money no matter which horse wins?
I found this question from Horse Betting Odds - But Guaranteed Win! and made some revision.
I think if I bet my money on A, B, C, and D under the following scenario, such as A:B:C:D = 20:15:12:10. I can always make money. These numbers come from the ratios 1/3 : 1/4 : 1/5 : 1/6 multiplied by 60， where 60 is the lowest common multiple for 3,4,5, and 6.
However, I wonder if there is a formal mathematical way to derive this solution. Is this question related to solving the linear inequality system or the probability?
probability algebra-precalculus systems-of-equations linear-programming
probability algebra-precalculus systems-of-equations linear-programming
edited Jan 4 at 11:34
user1551
71.8k566125
asked Jan 4 at 9:07
Bratt SwanBratt Swan
1234
edited Jan 4 at 11:34
user1551
71.8k566125
asked Jan 4 at 9:07
Bratt SwanBratt Swan
1234
edited Jan 4 at 11:34
user1551
71.8k566125
edited Jan 4 at 11:34
user1551
71.8k566125
edited Jan 4 at 11:34
user1551
71.8k566125
71.8k566125
asked Jan 4 at 9:07
Bratt SwanBratt Swan
1234
asked Jan 4 at 9:07
Bratt SwanBratt Swan
1234
asked Jan 4 at 9:07
Bratt SwanBratt Swan
1234
1234
add a comment |
add a comment |
1 Answer
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Each of the horses has a certain return when they win, A is a $6times$ return, B is $5times$, C $4times$, and D $3times$. In general say you have a set of horses $H_i$ that each have a return $r_i$, and you bet some total amount $B$ with $B_i$ on each horse $H_i$. As long as $B_i ge frac{B}{r_i}$ for all $i$, you will not lose money. That is because one of the $H_i$ will win, and that bet will give you $B_ir_i ge B$, so you end up with at least as much as you started. To maximize your guaranteed winnings, you want to maximize the minimum $B_ir_i$, as $B_ir_i-B$ is your net profit. This will happen when each $B_ir_i$ is equal, which occurs when
$$B_i = frac{frac{B}{r_i}}{sum frac{1}{r_i}}$$
and your guaranteed profit is
$$frac{B}{sum frac{1}{r_i}} - B$$
This is why in a real horse race $sum frac{1}{r_i}$ will always be greater than $1$ (if for some reason it's not go make a big bet!). In this question it was only $.95$ allowing you to make a nice little profit!
answered Jan 4 at 9:42
Erik ParkinsonErik Parkinson
9159
New contributor
Erik Parkinson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thanks for your explanation. One more question is that why when each $B_ir_i$ is equal, then the profit is maximized?
– Bratt Swan
Jan 4 at 10:51
@BrattSwan If they are not equal, say $B_ir_i >B_jr_j$, then you make more money if $H_i$ wins and less if $H_j$ does. So to maximize your guaranteed winnings no matter what the outcome it, you would want to move some of that $B_i$ money into $B_j$.
– Erik Parkinson
Jan 4 at 11:07
1
I.e. this maximises the minimum possible return for a given total bet.
– Henry
Jan 4 at 23:14
add a comment |
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Each of the horses has a certain return when they win, A is a $6times$ return, B is $5times$, C $4times$, and D $3times$. In general say you have a set of horses $H_i$ that each have a return $r_i$, and you bet some total amount $B$ with $B_i$ on each horse $H_i$. As long as $B_i ge frac{B}{r_i}$ for all $i$, you will not lose money. That is because one of the $H_i$ will win, and that bet will give you $B_ir_i ge B$, so you end up with at least as much as you started. To maximize your guaranteed winnings, you want to maximize the minimum $B_ir_i$, as $B_ir_i-B$ is your net profit. This will happen when each $B_ir_i$ is equal, which occurs when
$$B_i = frac{frac{B}{r_i}}{sum frac{1}{r_i}}$$
and your guaranteed profit is
$$frac{B}{sum frac{1}{r_i}} - B$$
This is why in a real horse race $sum frac{1}{r_i}$ will always be greater than $1$ (if for some reason it's not go make a big bet!). In this question it was only $.95$ allowing you to make a nice little profit!
answered Jan 4 at 9:42
Erik ParkinsonErik Parkinson
9159
New contributor
Erik Parkinson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thanks for your explanation. One more question is that why when each $B_ir_i$ is equal, then the profit is maximized?
– Bratt Swan
Jan 4 at 10:51
@BrattSwan If they are not equal, say $B_ir_i >B_jr_j$, then you make more money if $H_i$ wins and less if $H_j$ does. So to maximize your guaranteed winnings no matter what the outcome it, you would want to move some of that $B_i$ money into $B_j$.
– Erik Parkinson
Jan 4 at 11:07
1
I.e. this maximises the minimum possible return for a given total bet.
– Henry
Jan 4 at 23:14
add a comment |
Each of the horses has a certain return when they win, A is a $6times$ return, B is $5times$, C $4times$, and D $3times$. In general say you have a set of horses $H_i$ that each have a return $r_i$, and you bet some total amount $B$ with $B_i$ on each horse $H_i$. As long as $B_i ge frac{B}{r_i}$ for all $i$, you will not lose money. That is because one of the $H_i$ will win, and that bet will give you $B_ir_i ge B$, so you end up with at least as much as you started. To maximize your guaranteed winnings, you want to maximize the minimum $B_ir_i$, as $B_ir_i-B$ is your net profit. This will happen when each $B_ir_i$ is equal, which occurs when
$$B_i = frac{frac{B}{r_i}}{sum frac{1}{r_i}}$$
and your guaranteed profit is
$$frac{B}{sum frac{1}{r_i}} - B$$
This is why in a real horse race $sum frac{1}{r_i}$ will always be greater than $1$ (if for some reason it's not go make a big bet!). In this question it was only $.95$ allowing you to make a nice little profit!
answered Jan 4 at 9:42
Erik ParkinsonErik Parkinson
9159
New contributor
Erik Parkinson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thanks for your explanation. One more question is that why when each $B_ir_i$ is equal, then the profit is maximized?
– Bratt Swan
Jan 4 at 10:51
@BrattSwan If they are not equal, say $B_ir_i >B_jr_j$, then you make more money if $H_i$ wins and less if $H_j$ does. So to maximize your guaranteed winnings no matter what the outcome it, you would want to move some of that $B_i$ money into $B_j$.
– Erik Parkinson
Jan 4 at 11:07
1
I.e. this maximises the minimum possible return for a given total bet.
– Henry
Jan 4 at 23:14
add a comment |
Each of the horses has a certain return when they win, A is a $6times$ return, B is $5times$, C $4times$, and D $3times$. In general say you have a set of horses $H_i$ that each have a return $r_i$, and you bet some total amount $B$ with $B_i$ on each horse $H_i$. As long as $B_i ge frac{B}{r_i}$ for all $i$, you will not lose money. That is because one of the $H_i$ will win, and that bet will give you $B_ir_i ge B$, so you end up with at least as much as you started. To maximize your guaranteed winnings, you want to maximize the minimum $B_ir_i$, as $B_ir_i-B$ is your net profit. This will happen when each $B_ir_i$ is equal, which occurs when
$$B_i = frac{frac{B}{r_i}}{sum frac{1}{r_i}}$$
and your guaranteed profit is
$$frac{B}{sum frac{1}{r_i}} - B$$
This is why in a real horse race $sum frac{1}{r_i}$ will always be greater than $1$ (if for some reason it's not go make a big bet!). In this question it was only $.95$ allowing you to make a nice little profit!
answered Jan 4 at 9:42
Erik ParkinsonErik Parkinson
9159
New contributor
Erik Parkinson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Each of the horses has a certain return when they win, A is a $6times$ return, B is $5times$, C $4times$, and D $3times$. In general say you have a set of horses $H_i$ that each have a return $r_i$, and you bet some total amount $B$ with $B_i$ on each horse $H_i$. As long as $B_i ge frac{B}{r_i}$ for all $i$, you will not lose money. That is because one of the $H_i$ will win, and that bet will give you $B_ir_i ge B$, so you end up with at least as much as you started. To maximize your guaranteed winnings, you want to maximize the minimum $B_ir_i$, as $B_ir_i-B$ is your net profit. This will happen when each $B_ir_i$ is equal, which occurs when
$$B_i = frac{frac{B}{r_i}}{sum frac{1}{r_i}}$$
and your guaranteed profit is
$$frac{B}{sum frac{1}{r_i}} - B$$
This is why in a real horse race $sum frac{1}{r_i}$ will always be greater than $1$ (if for some reason it's not go make a big bet!). In this question it was only $.95$ allowing you to make a nice little profit!
answered Jan 4 at 9:42
Erik ParkinsonErik Parkinson
9159
New contributor
Erik Parkinson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Jan 4 at 9:42
Erik ParkinsonErik Parkinson
9159
New contributor
Erik Parkinson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Jan 4 at 9:42
Erik ParkinsonErik Parkinson
9159
answered Jan 4 at 9:42
Erik ParkinsonErik Parkinson
9159
9159
New contributor
Erik Parkinson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Erik Parkinson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Erik Parkinson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thanks for your explanation. One more question is that why when each $B_ir_i$ is equal, then the profit is maximized?
– Bratt Swan
Jan 4 at 10:51
@BrattSwan If they are not equal, say $B_ir_i >B_jr_j$, then you make more money if $H_i$ wins and less if $H_j$ does. So to maximize your guaranteed winnings no matter what the outcome it, you would want to move some of that $B_i$ money into $B_j$.
– Erik Parkinson
Jan 4 at 11:07
1
I.e. this maximises the minimum possible return for a given total bet.
– Henry
Jan 4 at 23:14
add a comment |
Thanks for your explanation. One more question is that why when each $B_ir_i$ is equal, then the profit is maximized?
– Bratt Swan
Jan 4 at 10:51
@BrattSwan If they are not equal, say $B_ir_i >B_jr_j$, then you make more money if $H_i$ wins and less if $H_j$ does. So to maximize your guaranteed winnings no matter what the outcome it, you would want to move some of that $B_i$ money into $B_j$.
– Erik Parkinson
Jan 4 at 11:07
1
I.e. this maximises the minimum possible return for a given total bet.
– Henry
Jan 4 at 23:14
Thanks for your explanation. One more question is that why when each $B_ir_i$ is equal, then the profit is maximized?
– Bratt Swan
Jan 4 at 10:51
Thanks for your explanation. One more question is that why when each $B_ir_i$ is equal, then the profit is maximized?
– Bratt Swan
Jan 4 at 10:51
@BrattSwan If they are not equal, say $B_ir_i >B_jr_j$, then you make more money if $H_i$ wins and less if $H_j$ does. So to maximize your guaranteed winnings no matter what the outcome it, you would want to move some of that $B_i$ money into $B_j$.
– Erik Parkinson
Jan 4 at 11:07
@BrattSwan If they are not equal, say $B_ir_i >B_jr_j$, then you make more money if $H_i$ wins and less if $H_j$ does. So to maximize your guaranteed winnings no matter what the outcome it, you would want to move some of that $B_i$ money into $B_j$.
– Erik Parkinson
Jan 4 at 11:07
1
1
I.e. this maximises the minimum possible return for a given total bet.
– Henry
Jan 4 at 23:14
I.e. this maximises the minimum possible return for a given total bet.
– Henry
Jan 4 at 23:14
add a comment |
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