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Cacarulo: An interesting problem
> Assuming the numbers below are correct, the
> probability is
> \Phi(Sqrt(200)*0.08417 / 1.314) = 81.75%
> Still far from certain.
would be to find "n" where the probability is 50%.
Sincerely,
Cacarulo
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Geoff Hall: Re: the probability is
> Assuming the numbers below are correct, the
> probability is
> \Phi(Sqrt(200)*0.08417 / 1.314) = 81.75%
> Still far from certain.
> Regards,
> Karel
Thanks Karel - seems a nice proposition at even money though.
What does Phi mean ?
Best regards
Geoff
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Geoff Hall: Re: An interesting problem
> would be to find "n" where the
> probability is 50%.
> Sincerely,
> Cacarulo
I may have misinterpreted this but surely you could not have a 50% situation as 'hitting' is superior to 'standing'.
In other words, after just 1 hand would be the closest that 'standing' comes to 50% chance of winning and this would get smaller as the number of trials increased.
P(Hitting Wins Overall)>50%>P(Standing Wins Overall)
Is this not correct ?
Yours confusingly
Geoff
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Cacarulo: Re: An interesting problem
> I may have misinterpreted this but surely
> you could not have a 50% situation as
> 'hitting' is superior to 'standing'.
> In other words, after just 1 hand would be
> the closest that 'standing' comes to 50%
> chance of winning and this would get smaller
> as the number of trials increased.
> P(Hitting Wins Overall)>50%>P(Standing
> Wins Overall)
> Is this not correct ?
You're right. This is not the appropiate example.
Sincerely,
Cacarulo
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Karel: Re: the probability is
See my other post. The proposition may not be that great any more, given the large bet. Depends on wealth and risk aversion.
$\Phi(t)$ is the cumulative normals distribution, the integral from $-\infty$ to $t$ of normal density $1/Sqrt(2\Pi) Exp(-x^2/2)$.
Anyway, if this should take place, I would not mind to share PART of such a bet. :-)
Regards,
Karel
> Thanks Karel - seems a nice proposition at
> even money though.
> What does Phi mean ?
> Best regards
> Geoff
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Karel: For general $p$
> would be to find "n" where the
> probability is 50%.
This $n$ is one for 50%, of course For a general probability $p$ we have the equation
\Phi(Sqrt(n)*0.08417 / 1.314) = p
The solution to this equation is given by the quantile $\Phi^{-1}$ of normal distribution:
n = (1.314/0.08417 * \Phi^{-1}(p) )^2
In Excel, \Phi^{-1}(p) can be written as
NORMSINV(p)
For example, for p=95% we get
n = 659.37,
so 660 games to be certain with probability 95%.
Regards,
Karel
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Don Schlesinger: Re: Not necessarily
> From the calculation below, the edge is
> roughly 60%. Given the huge bet size, the
> certainty equivalent may well be close to
> zero for reasonable risk aversion
> coefficients and reasonable wealth size.
"Reasonable wealth" meaning not so much, right?
> For example, with probability of winning of
> 80% (edge 60%), bet size of 5% of total
> wealth, and risk aversion $p=30$ (based on
> total investment wealth, thus not too risk
> averse),
What does "$p=30$" mean???
> the certainty equivalent is already
> negative 5.27%. Thus, the player would be
> willing to PAY in order to avoid such a bet
Send him to me! I'll gladly accept the wager, at even money, with 4 to 1 odds in my favor, for $15,000.
Don
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Don Schlesinger: Just change the number of hands
A suggestion: this bet involves an intelligent person and a moron. I don't think 200 hands should be sacrosanct. If this person believes standing is superior to hitting, then surely, he'd like the experiment to display the truth, right?
After all, he wouldn't want to play just 10 hands, and risk being proven wrong by a fluke occurrence, would he?
I guess he won't trust a book or computer sim, so we're going to have to deal hands. But, for $15,000, why stop at 200? Who's in a rush? Let's get it right. So, let's deal 500 or 600 hands ... or more. What's the difference? If the moron thinks he's got the right side of things, he should want MORE hands, not fewer, right?
Just a suggestion.
Don
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Karel: Of course...
> "Reasonable wealth" meaning not so
> much, right?
Yes, meaning not so much. The calculation was based on total wealth, including everything (income, etc.) of only $200,000. This is related to my economics paper about risk aversion. I am very modest there since I try to make the point as strong as possible.
> What does "$p=30$" mean???
This is the risk aversion. $p=30$ means 1/30 Kelly, based on total wealth. Not really too conservative.
> Send him to me! I'll gladly accept the
> wager, at even money, with 4 to 1 odds in my
> favor, for $15,000.
Of course, the "reasonable" wealth assumption is not valid in this case :-)
Anyway, I mentioned the result also as proportion of total wealth. Would you like to participate in this bet, if the bet size was 5% of all your wealth, take it or leave it?
Regards,
Karel
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Geoff Hall: Re: Just change the number of hands
> A suggestion: this bet involves an
> intelligent person and a moron. I don't
> think 200 hands should be sacrosanct. If
> this person believes standing is superior to
> hitting, then surely, he'd like the
> experiment to display the truth, right?
> After all, he wouldn't want to play just 10
> hands, and risk being proven wrong by a
> fluke occurrence, would he?
> I guess he won't trust a book or computer
> sim, so we're going to have to deal hands.
> But, for $15,000, why stop at 200? Who's in
> a rush? Let's get it right. So, let's deal
> 500 or 600 hands ... or more. What's the
> difference? If the moron thinks he's got the
> right side of things, he should want MORE
> hands, not fewer, right?
> Just a suggestion.
> Don
Good point although generally when dealing with this type of person they fail to understand any form of logic.
I noticed that Karel said that 660 hands would give a 95% degree of confidence so I will suggest that my friend tries to persuade this player to deal 700 hands using your 'logic' to him.
I will post the outcome if ever the bet takes place.
Thanks to all those who posted their analysis and suggestions.
Best regards
Geoff
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Don Schlesinger: Re: Of course...
> Anyway, I mentioned the result also as
> proportion of total wealth. Would you like
> to participate in this bet, if the bet size
> was 5% of all your wealth, take it or leave
> it?
Interesting question. I'd be very tempted to accept (but my wife would probably kill me! :-)). I'll have to think about it. Good question!
Don
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Don Schlesinger: Wife shocked me! :-)
Well, I'm very proud of my wife! She thought about it long and hard. Wasn't a clearcut choice at all.
I was quite shocked, to tell the truth. Finally, she did admit, "That's a lot of money," but she didn't say it as if it would be unthinkable to make the wager.
As for me, after losing about 20-25% of my entire net worth in the stock market these past three years, I think another 5% wouldn't kill me.
I'd make the bet, Karel.
Now, can you show me the math (which I probably won't understand) as to why it's wrong to do so, please.
Don
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Susan: Re: Just change the number of hands
...
> persuade this player to deal 700 hands using
> your 'logic' to him.
> I will post the outcome if ever the bet
> takes place.
> Thanks to all those who posted their
> analysis and suggestions.
> Best regards
> Geoff
Go for 500 or even better 1000 hands. He may think you are trying to trick him into something unusual with 700 hands.
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