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Primes and the Geometric Distribution 
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#1
Jun704, 04:21 PM

P: 20

Given the probability of flipping a heads with a fair coin is [tex]\frac{1}{2}[/tex], what is the probability that the first heads occurs on a prime number?



#2
Jun704, 04:59 PM

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PF Gold
P: 2,226

[tex]\sum_{n=1}^\infty \left(\frac{1}{2}\right)^{p_n}[/tex]
where [itex]p_n[/itex] is the nth prime number. Which gives a value of about 0.41468 


#3
Jun704, 04:59 PM

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HW Helper
P: 2,586

Let {[itex]p_i[/itex]} be the sequence of primes. The probability you're looking for would be:
[tex]\sum _{i=1} ^{\infty} 0.5^{p_i}[/tex] 


#4
Jun704, 05:07 PM

P: 20

Primes and the Geometric Distribution
Yes, I realize that the answer is this summation ([tex]\sum _{primes} ^{} 1/2^p[/tex]), which clearly converges very quickly (to about .4146825...). Anyone have any ideas on whether the answer is irrational, or even expressable as a fraction of constants (like [tex]\sum _{n=1} ^{\infty} 1/n^2[/tex]. Anyone know anything else about [tex]f(x)=\sum _{primes} ^{} 1/x^p[/tex]?



#5
Jun704, 05:17 PM

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PF Gold
P: 2,226

I don't really know what more you want, if we wee able to ask your question precisely we'd be to busy polishing our Fields medals to post on Physics Forums.
We can say the series is convergant, but I don't think there's too much more we can say. 


#6
Jun704, 05:31 PM

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PF Gold
P: 2,226

There are alot of products and sums involving the nth primes that converge very quickly to a value, there are even somewhoe precise value can be known, but I'm pretty ceratin that this isn't one of them.



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