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  #11  
Old 04-17-2012, 11:59 PM
ivanku ivanku is offline
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Default Re: question about probability

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Originally Posted by yaser View Post
This is approximately {1 \over e} since \lim_{n\to\infty} (1-{1\over n})^n = {1\over e}. Numerically, {1 \over e}\approx{1\over 2.718}\approx 0.37.
Is it correct to assume that \nu_{min} from the coins tossing simulation in homework 2 assignment one should be close to that limit (where we're tossing 1000 coins 10 times and \nu_{min} is the fraction of heads obtained for the coin which had the minimum frequency of heads)?
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  #12  
Old 04-18-2012, 02:49 AM
elkka elkka is offline
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Default Re: question about probability

The law of big numbers states that the average $\nu_min$ is close to the $E{\nu_min}$.

$E\nu_min$ can be calculated directly for this experiment.
$P(\nu_min=0)$=0.623576
$P(\nu_min = 0.1)$ = 0.3764034
$P(\nu_min = 0.2)$ = 0.00002;
and $P(\nu_min>=0.3)=0$ for the purposes of calculating the mean.

Therefore, $E(\nu_min)$=0.037644, and the average proportion of heads for c_min should be close to this number.
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  #13  
Old 04-18-2012, 05:43 AM
SamK52 SamK52 is offline
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Default Re: question about probability

Allow me to format your post:

Quote:
The law of big numbers states that the average \nu_{min} is close to the E{\nu_{min}}.

E\nu_{min} can be calculated directly for this experiment.
P(\nu_{min}=0)=0.623576
P(\nu_{min} = 0.1)= 0.3764034
P(\nu_{min} = 0.2) = 0.00002
and P(\nu_{min}>=0.3)=0 for the purposes of calculating the mean.

Therefore, E(\nu_{min})=0.037644, and the average proportion of heads for c_{min} should be close to this number.
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  #14  
Old 04-18-2012, 10:26 AM
elkka elkka is offline
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Default Re: question about probability

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Originally Posted by SamK52 View Post
Allow me to format your post:
Please, allow me to ask how you did it?
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  #15  
Old 04-18-2012, 12:57 PM
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yaser yaser is offline
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Default Re: question about probability

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Originally Posted by elkka View Post
Please, allow me to ask how you did it?
http://book.caltech.edu/bookforum/sh...77&postcount=1
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  #16  
Old 04-18-2012, 12:58 PM
rohanag rohanag is offline
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Default Re: question about probability

how did you calculate those probability values? ( nu_min = 0, 0.1, 0.2 )
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  #17  
Old 04-19-2012, 03:10 AM
elkka elkka is offline
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Default Re: question about probability

Thank you, Professor.

This how I calculate the probabilities. Let h_{min} = 10*\nu_{min} - the number of heads for c_{min}, and let h_i be the number of heads in i-th experiment (out of 1000). Then, as Professor has shown previously,

P(\nu_{min}=0) =P(h_{min}=0) = P(\exists i\in[1,1000]\,\, h_i = 0)=1-P(\forall i\in[1,1000]\, h_i > 0)
=1-P(h_1 > 0)^{1000};

Now, P(h_1>0) = 1-P(h_1 = 0) = 1-0.5^{10} . Therefore,
P(\nu_{min}=0)=1-(1-0.5^{10})^{1000} = 0.623576.

Next,
P(\nu_min=0.1)=P(h_{min}=1) =P(\forall i\in[1,1000] \,\,h_i > 0)-P(\forall i\in[1,1000] \,\,h_i > 1)
= P^{1000}(h_1>0)-P^{1000}(h_1>1) = (1-P(h_1=0))^{1000}-(1-P(h_1=0)-P(h_1=1))^{1000}
=(1-0.5^{10})^{1000}-(1-0.5^{10}-C_{10}^1*0.5^{10})^{1000} =0.3764034.

Next,
P(\nu_min=0.2)=P(h_{min}=2) =P(\forall i\in[1,1000]\,\, h_i > 1)-P(\forall i\in[1,1000] \,\,h_i > 2)
= P^{1000}(h_1>1)-P^{1000}(h_1>2)
= (1-P(h_1=0)-P(h_1=1))^{1000}-(1-P(h_1=0)-P(h_1=1)-P(h_1=2))^{1000}
=(1-0.5^{10}-C_{10}^1*0.5^{10})^{1000}-(1-0.5^{10}-C_{10}^1*0.5^{10}-C_{10}^2*0.5^{10})^{1000} =0.000204.

The rest can be calculated directly too, but they are essenctially 0 for the purpose of calculating the mean.
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  #18  
Old 04-19-2012, 07:44 AM
rohanag rohanag is offline
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Default Re: question about probability

thank you for the detailed explanation.
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  #19  
Old 07-13-2012, 04:55 PM
nyxee nyxee is offline
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Default Re: question about probability

the answer is very clear but how do we know when to use this not.

to clarify my question, if say P(ten heads)=p and P(not ten heads)=q (=1-p). why does using (p^1000) give the wrong answer?
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  #20  
Old 07-15-2012, 03:56 AM
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htlin htlin is offline
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Default Re: question about probability

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Originally Posted by nyxee View Post
the answer is very clear but how do we know when to use this not.

to clarify my question, if say P(ten heads)=p and P(not ten heads)=q (=1-p). why does using (p^1000) give the wrong answer?
p^{1000} means the probability of getting ten heads in each of the independent random trials. Is that the event you are interested in?
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