## RE: [PrimeNumbers] odds being prime in a bar

Expand Messages
• ... I was thinking more in these lines for the barhavioral science problem. There are 4 primes
Message 1 of 1 , Jan 31, 2004
• 0 Attachment
>To: "'cino hilliard'" <hillcino368@...>,
>CC: <miltbrown@...>
>Subject: RE: [PrimeNumbers] odd being prime in a bar
>Date: Thu, 29 Jan 2004 21:23:39 -0800
>
>There are infinitely many more odd primes than even primes!
>

I was thinking more in these lines for the barhavioral science problem.
There are 4 primes < 10 so if
the group picks numbers less than= 10 they will on average pick 4 prime
numbers. So the odds of
picking a prime is 4/10. Then the odds of picking 10 primes is (4/10)^10 =
.000105. The odds of
10 primes for numbers < 100 = (25/100)^10 = 0.00000095367431. For numvers <
1000 odds =
0.000000017909885 etc.

The Pari program verifies this.

\\n = number persons asked, maxpow10 = max pow 10 they will write, trials =
number of
\\times we ask different sets of n persons. So probability is
\\total number of primes found / total writes = primesfound/(trials*n)
oddsprime(n,maxpow10,trials) =
{
default(realprecision,14);
for(p=1,maxpow10,
primesfound=0;
for(j=1,trials,
c=0;
for(x=1,n,
y=random(10^p);
if(isprime(y),primesfound++)
);
);
print(n" 10^"p" "primesfound" "primesfound/trials/n+.0"
"pi(10^p))
);
default(realprecision,28);
}

pi(n) =
{
c=0;
forprime(x=2,n,c++);
return(c)
}

Some output
10 10^1 801197 0.40059850000000 4
10 10^2 500063 0.25003150000000 25
10 10^3 335655 0.16782750000000 168
10 10^4 245158 0.12257900000000 1229
10 10^5 191652 0.095826000000000 9592
10 10^6 157038 0.078519000000000 78498
10 10^7 132894 0.066447000000000 664579

So the odds of getting 10 primes are (pi(max num to write)/(max no to
write))^10

So if we assume the group will pick numbers up to 10^22 then
the odds of hitting 10 primes is (201467286689315906290/10^22)^10 =
1.101654404308712330002053882 E-17

For larger numbers we could use the logrithmic integral Li(x) =
-eint1(log(1/x)) in pari.

In this example Li(10^22) = 201467286691248261498.1505283 the odds =
(Li(10^22)/10^22)^10 = 1.101654404414376513047443655 E-17 this differs from
the pi(x)
calculation by 1.0566418304538977300 E-27.

Now that is an unreasonable assumption for a group of bar folks. My
reasonable guess would be
these folks will stay in the less than= 100 range for a
0.00000095367431640625 prob.

Indeed (Li(10^100)/10^100)^10 = 2.493611096381172592960013489 E-24
It is doubtful we will ever find pi(10^100) so Li(10^100) =
4.361971987140703159099509112 E97
will have to suffice. Maybe an indistructable computer that starts today and
a time machine? We go into the future a few 10^n years, get the answer and
come back with it.

Cino
"Behavior is not for the pursuit of survival but because of it."

>-----Original Message-----
>From: cino hilliard [mailto:hillcino368@...]
>Sent: Thursday, January 29, 2004 8:57 PM
>Subject: Re: [PrimeNumbers] odd being prime in a bar
>
>Hi,
>This may at first seem as "unsolvable" but what would be a reasonable
>solution.
>
>Ten people in a bar are asked to write down a number. What are the odds
>that
>all ten
>will pick a prime number?
>
>How about Twenty people? Does reasonableness improve as the number of
>people
>increase?
>
>I think this can be solved analytically within reason of course.
>
>Cino
>

_________________________________________________________________
There are now three new levels of MSN Hotmail Extra Storage! Learn more.
http://join.msn.com/?pgmarket=en-us&page=hotmail/es2&ST=1
Your message has been successfully submitted and would be delivered to recipients shortly.