Frequency of the Prime Numbers

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Suppose I took all natural numbers less than or equal to $x$ and I picked one at random. Is there a way that we know of to express the probability that my number is prime in terms of $x$, for all $x$?



For example, for $x=12$, the prime numbers less than or equal to $x$ are $2,3,5,7$ and $11$, so my probability is $5/12$.




probability number-theory prime-numbers




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    up vote
    5
    down vote

    favorite












    Suppose I took all natural numbers less than or equal to $x$ and I picked one at random. Is there a way that we know of to express the probability that my number is prime in terms of $x$, for all $x$?



    For example, for $x=12$, the prime numbers less than or equal to $x$ are $2,3,5,7$ and $11$, so my probability is $5/12$.




    probability number-theory prime-numbers




    share|cite|improve this question
























      up vote
      5
      down vote

      favorite









      up vote
      5
      down vote

      favorite











      Suppose I took all natural numbers less than or equal to $x$ and I picked one at random. Is there a way that we know of to express the probability that my number is prime in terms of $x$, for all $x$?



      For example, for $x=12$, the prime numbers less than or equal to $x$ are $2,3,5,7$ and $11$, so my probability is $5/12$.




      probability number-theory prime-numbers




      share|cite|improve this question














      Suppose I took all natural numbers less than or equal to $x$ and I picked one at random. Is there a way that we know of to express the probability that my number is prime in terms of $x$, for all $x$?



      For example, for $x=12$, the prime numbers less than or equal to $x$ are $2,3,5,7$ and $11$, so my probability is $5/12$.





      probability number-theory prime-numbers





      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Aug 8 at 10:33









      Jam

      4,35811330




      4,35811330










      asked Aug 8 at 10:27









      Tsing Shi Tao

      444




      444




















          1 Answer
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          up vote
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          There is no explicit formula, but the prime number theorem says that as $x$ tends to infinity, there are around $fracxln x$ primes $≤ x$.



          This means that the chance that a random number is prime will be around $fracxln x cdot frac1x = frac1ln x$.






          share|cite|improve this answer
















          • 4




            In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
            – infinitezero
            Aug 8 at 12:34










          • infinity is a bit further
            – dEmigOd
            Aug 8 at 12:35






          • 1




            @infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
            – Toby Mak
            Aug 8 at 12:36











          • I know. I just pointed out that even for small X it's pretty close.
            – infinitezero
            Aug 8 at 12:37






          • 5




            $frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
            – probably_someone
            Aug 8 at 13:22











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          1 Answer
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          active

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          active

          oldest

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          active

          oldest

          votes








          up vote
          23
          down vote













          There is no explicit formula, but the prime number theorem says that as $x$ tends to infinity, there are around $fracxln x$ primes $≤ x$.



          This means that the chance that a random number is prime will be around $fracxln x cdot frac1x = frac1ln x$.






          share|cite|improve this answer
















          • 4




            In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
            – infinitezero
            Aug 8 at 12:34










          • infinity is a bit further
            – dEmigOd
            Aug 8 at 12:35






          • 1




            @infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
            – Toby Mak
            Aug 8 at 12:36











          • I know. I just pointed out that even for small X it's pretty close.
            – infinitezero
            Aug 8 at 12:37






          • 5




            $frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
            – probably_someone
            Aug 8 at 13:22















          up vote
          23
          down vote













          There is no explicit formula, but the prime number theorem says that as $x$ tends to infinity, there are around $fracxln x$ primes $≤ x$.



          This means that the chance that a random number is prime will be around $fracxln x cdot frac1x = frac1ln x$.






          share|cite|improve this answer
















          • 4




            In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
            – infinitezero
            Aug 8 at 12:34










          • infinity is a bit further
            – dEmigOd
            Aug 8 at 12:35






          • 1




            @infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
            – Toby Mak
            Aug 8 at 12:36











          • I know. I just pointed out that even for small X it's pretty close.
            – infinitezero
            Aug 8 at 12:37






          • 5




            $frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
            – probably_someone
            Aug 8 at 13:22













          up vote
          23
          down vote










          up vote
          23
          down vote









          There is no explicit formula, but the prime number theorem says that as $x$ tends to infinity, there are around $fracxln x$ primes $≤ x$.



          This means that the chance that a random number is prime will be around $fracxln x cdot frac1x = frac1ln x$.






          share|cite|improve this answer












          There is no explicit formula, but the prime number theorem says that as $x$ tends to infinity, there are around $fracxln x$ primes $≤ x$.



          This means that the chance that a random number is prime will be around $fracxln x cdot frac1x = frac1ln x$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 8 at 10:29









          Toby Mak

          2,8051925




          2,8051925







          • 4




            In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
            – infinitezero
            Aug 8 at 12:34










          • infinity is a bit further
            – dEmigOd
            Aug 8 at 12:35






          • 1




            @infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
            – Toby Mak
            Aug 8 at 12:36











          • I know. I just pointed out that even for small X it's pretty close.
            – infinitezero
            Aug 8 at 12:37






          • 5




            $frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
            – probably_someone
            Aug 8 at 13:22













          • 4




            In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
            – infinitezero
            Aug 8 at 12:34










          • infinity is a bit further
            – dEmigOd
            Aug 8 at 12:35






          • 1




            @infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
            – Toby Mak
            Aug 8 at 12:36











          • I know. I just pointed out that even for small X it's pretty close.
            – infinitezero
            Aug 8 at 12:37






          • 5




            $frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
            – probably_someone
            Aug 8 at 13:22








          4




          4




          In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
          – infinitezero
          Aug 8 at 12:34




          In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
          – infinitezero
          Aug 8 at 12:34












          infinity is a bit further
          – dEmigOd
          Aug 8 at 12:35




          infinity is a bit further
          – dEmigOd
          Aug 8 at 12:35




          1




          1




          @infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
          – Toby Mak
          Aug 8 at 12:36





          @infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
          – Toby Mak
          Aug 8 at 12:36













          I know. I just pointed out that even for small X it's pretty close.
          – infinitezero
          Aug 8 at 12:37




          I know. I just pointed out that even for small X it's pretty close.
          – infinitezero
          Aug 8 at 12:37




          5




          5




          $frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
          – probably_someone
          Aug 8 at 13:22





          $frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
          – probably_someone
          Aug 8 at 13:22













           

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