Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Primes and the Geometric Distribution

  1. Jun 7, 2004 #1

    CTS

    User Avatar

    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. jcsd
  3. Jun 7, 2004 #2

    jcsd

    User Avatar
    Science Advisor
    Gold Member

    [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
     
    Last edited: Jun 7, 2004
  4. Jun 7, 2004 #3

    AKG

    User Avatar
    Science Advisor
    Homework Helper

    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]
     
  5. Jun 7, 2004 #4

    CTS

    User Avatar

    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]?
     
  6. Jun 7, 2004 #5

    jcsd

    User Avatar
    Science Advisor
    Gold Member

    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.
     
  7. Jun 7, 2004 #6

    jcsd

    User Avatar
    Science Advisor
    Gold Member

    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.
     
  8. Jun 9, 2004 #7

    Gokul43201

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    If you had posted this question several centuries ago, I might have said:

    "the sum approaches [tex]\sqrt{2} - 1 [/tex] but there is not enough room in this forum for me to prove it. "

    and gotten away with that !
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Primes and the Geometric Distribution
  1. Geometric Distribution (Replies: 3)

Loading...