Every soul is allotted 6 numbers, chosen at random from the positive integers.
If such numbers are relatively prime, then the soul is admitted to Heaven - otherwise, not.
What is the probability of an eternal reward?
What is the probability of an eternal reward, if such numbers must be pairwise prime?
Note: the integers 30, 42, 70, and 105 are relatively prime, while the numbers 121, 122, and 123 are pairwise prime, see the definitions at https://primes.utm.edu/glossary/xpage/RelativelyPrime.html
I had heard of the simpler problem before:
The probability that two numbers are relatively prime is 1/zeta(2) = 6/pi^2=0.60793
This link shows how to generalize the first question
https://mathworld.wolfram.com/RelativelyPrime.html
For n numbers the solution is 1/zeta(n)
Here are some exact values of the zeta function
https://mathworld.wolfram.com/RiemannZetaFunction.html
From which the solution to the first question is 1/zeta(6)=945/pi^6=0.98295
(This is pretty high, which is unsurprising, as we'd only be barred from heaven if all 6 numbers are even or all multiples of 3, etc.)
For the second part we have to consider each of the C(6,2)=15 pairs. (This is not good for out eternal souls. Now if any two of the six numbers are even etc. we are damned.)
For the probability, assuming the numbers were chosen independently, just do
(6/pi^2)^15=0.00057250
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Posted by Jer
on 2021-07-01 10:36:27 |
Solution with research
