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Thread: It's A Horse Race

20040403, 03:09 #1
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It's A Horse Race
You've just received a 100% reliable tip: You know with absolute certainty that one of three horses would win the sixth race.
Greased Lightning, a 21 shot; Honey Bear, a 31 shot; or Splendid Wind, a 41 shot.
How much would you have to bet on each of the three horses to be assured of coming out with a $39 profit? Ricky

20040403, 18:54 #2
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Re: It's A Horse Race
Either I misunderstand your question or the odds are misquoted. There is no way to be assured that you can come out with (at least) a $39 profit:
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>Assume you bet A dollars on Greased Lightning, B Dollars on Honey Bear and C Dollars on Splendid Wind.
You have Spent A+B+C dollars. If Greased Lightning wins you win 2A Dollars (and lose A+B+C) so that you have to ensure that:
2AABC>=39
ABC>=39
A>=39+B+C
If Honey Bear Wins, again you spend A+B+C, but you win 3B so you have to ensure that:
3BABC>=39
2BAC>=39
2B  (39+B+C) C >=39
B392C >= 39
B>=78+2C
If Splendid WInd WIns you have to ensure that:
4CABC>=39
3CAB>=39
3C(39+B+C)  (78+2C) >=39
3C  39  B  C78 2C >=39
B >= 156
(78+2C) >=156
78 2C >=156
2C >=234
2C< 234
C<234/2
C<117
So to make a 39$ profit, You must bet at least $117 on Splendid wind, and with that bet you would bet $234 on Greased lightning and $156 on Honey Bear.
This means to be assured $39 profit on this "sure thing" you must be a bookie and ensure that some else bets $234 on Greased lightning, $156 on Honey Bear and $117 on Splendid wind.</font color=yellow></span hi>
Steve

20040404, 01:30 #3
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Re: It's A Horse Race
Okay, here's a starting place...
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>First you must translate the odds into probabilities. Note that the chance of Greased Lightning winning is 1/3; similarly, Honey Bear has a 1/4 chance of winning, and Splendid Wind a 1/5 chance. </font color=yellow></span hi> Ricky

20040404, 11:49 #4
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Re: It's A Horse Race
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>It doesn't matter what their "Probabilities of winning are". One of three things will happen:
Horse1 will win and I must make (at least) $39 profit
Horse2 will win and I must make (at least) $39 profit
or Horse3 will win and I must make (at least) $39 profit
You said we must make at least $39 profit no matter how we bet. The odds give you the payout. You didn't say we needed to have a "probability of making $39", or am I "interpreting" the puzzle wrong:
<hr>How much would you have to bet on each of the three horses to be assured of coming out with a $39 profit?<hr>
This states to me, I put some money on each of the horses. No matter which one wins, I will make (at least) $39. It is not a "probability of profit" it is an "assurance of profit", a "sure thing", I will not lose money in this bet. I don't think that is possible unless you take the bets as I previously determined.</font color=yellow></span hi>
I am curious to see your answer to show me my error (If I am mistaken), because I worked it many ways, and unless you are changing the puzzle or show me where I am misinterpreting, I stand by my answer.
Steve

20040404, 12:43 #5
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Re: It's A Horse Race
I can not find a solution to this one where the punter wins.
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>]In your original post you said it was 100% reliable tip and you know with absolute certainty that 1 of the 3 horses would win. Therefore the sum of probabilities of the 3 horses winning must be 100%, but adding your probabilites together comes to about 78%
Unless I'm missing something I agree with Steve</font color=yellow></span hi>

20040404, 16:04 #6
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Re: It's A Horse Race
I can't find a solution to this. If the profit has to be exactly $39, regardless of which of the 3 horses wins, all I can come up with is this, with A, B & C being the amounts bet on each respective horse:
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>
A wins: A  B  C =39
B wins: A + 2B  C = 39
C wins: A B + 3C = 39
And there ain't no solution with A,B,C > 0
Also, "odds" are a reflection of probabilities in horse races, not actual probabilities.
</font color=yellow></span hi>
Alan

20040404, 17:25 #7
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Re: It's A Horse Race
This is exactly how I approached it only I worked with >=39 since I assumed you could win $39 in one "case" and more than 39 in the other 2 cases. It just was that you had to ("assured") win at least $39.
There is not solution (I think) other than the other I proposed:
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow> you act as bookie and get someone else to bet the money: the "house" always wins</font color=yellow></span hi>
Steve

20040404, 18:42 #8
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Re: It's A Horse Race
Okay, like it or not, here's the official solution to the puzzle...
<hr>...Greased Lightning, a 21 shot; Honey Bear, a 31 shot; or Splendid Wind, a 41 shot. How much would you have to bet on each of the three horses to be assured of coming out with a $39 profit?<hr><span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>First we must translate the odds into probabilities. Note that the chance of Lightning winning is 1/3; similarly, Honey has a 1/4 chance of winning, and Splendid a 1/5 chance.
We seek amounts A,B, and C such that A + B + C + 39 = 3A = 4B = 5C. Therefore, B = (3/4)A and C = (3/5)A. Substituting gives (3/4)A + (3/5)A + 39 = 2A., so 39 = (13/20)A, so A = 60. B = 45. C = 36. In summary, we must bet $60 on Lightning, $45 on Honey and $36 on Splendid.
To follow through with an example, let's assume Honey is the winner. After you place your three bets, you have laid out $141. The winning bet on Honey would return $180 (that's 31 at $45). $180  $141 = $39</font color=yellow></span hi> Ricky

20040404, 20:35 #9
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Re: It's A Horse Race
I guess I am confused about "payout" vs "profit". It seems that I don't do enough betting <img src=/S/grin.gif border=0 alt=grin width=15 height=15>
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>My assumption was that 3:1 pays $3 for every $1 spent, so your profit is $2 for every dollar spent. You are saying that, my "assumption" is incorrect and why I got the wrong numbers. 3:1 means that you make $3 profit for every $1 you spend, your payout is actually $4 (3+1) for every $1 spent.</font color=yellow></span hi>
Steve

20040404, 22:56 #10
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Re: It's A Horse Race
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>I can see what you mean. If a payout is 31, it seems simple enough to believe that you'd get $3 in return for a $1 bet. But, what isn't obvious at first, is that you'd have your original wager returned as well. Consider an even money bet; 11. If you only got back exactly what you bet, then no one would bother. A 11 bet would actually return your original wager and a matching sum.</font color=yellow></span hi>
 Ricky

20040405, 00:06 #11
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Re: It's A Horse Race
It appears that my betting is pretty rusty too. As you say later, if a horse is "at evens" then it pays out at 1:1, but this means that you get your amount bet refunded plus 1 x that amount bet as your payout. This is rather embarassing <img src=/S/blush.gif border=0 alt=blush width=15 height=15> considering I used to have a dabble myself. More interesting is your comment:<hr>the "house" always wins<hr>I might as well toss this in as a puzzle here, but it's actually a true story of how a group of knowledgable punters legally ripped off the casino here to the tune of 7 figures. They exploited a weakness in the casino's system of getting/ giving odds, which involved having odds phoned in directly from the meets (the local track betting facility), rather than paying the official "totalizer" for the use of their facilities and figures. Someone with some knowledge of race meets and how odds are determined might hazard a guess as to how they turned it round in their favour, and made the "house" lose bigtime. Hint: They did it via small country meets.
Alan