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LFD Book Forum (http://book.caltech.edu/bookforum/index.php)
-   Chapter 1 - The Learning Problem (http://book.caltech.edu/bookforum/forumdisplay.php?f=108)
-   -   question about probability (http://book.caltech.edu/bookforum/showthread.php?t=287)

ivanku 04-18-2012 12:59 AM
Re: question about probability
 
Quote:
Originally Posted by yaser (Post 1288)
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)?

elkka 04-18-2012 03:49 AM
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.

SamK52 04-18-2012 06:43 AM
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.

elkka 04-18-2012 11:26 AM
Re: question about probability
 
Quote:
Originally Posted by SamK52 (Post 1399)
Allow me to format your post:
Please, allow me to ask how you did it?

yaser 04-18-2012 01:57 PM
Re: question about probability
 
Quote:
Originally Posted by elkka (Post 1413)
Please, allow me to ask how you did it?
http://book.caltech.edu/bookforum/sh...77&postcount=1

rohanag 04-18-2012 01:58 PM
Re: question about probability
 
how did you calculate those probability values? ( nu_min = 0, 0.1, 0.2 )

elkka 04-19-2012 04:10 AM
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.

rohanag 04-19-2012 08:44 AM
Re: question about probability
 
thank you for the detailed explanation.

nyxee 07-13-2012 05:55 PM
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?

htlin 07-15-2012 04:56 AM
Re: question about probability
 
Quote:
Originally Posted by nyxee (Post 3399)
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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