So the Goldbach conjecture says 'Every even integer greater than 2 can be written as sum of two primes'. Here is what I have roughly done to verify it, using probability. I don't say it is correct but I just want to show it.
Let a be an even integer greater than 2. The prime counting function gives number of primes n as n ~ a/ln a. Now, for every prime less than a, we generate odd numbers(except for 2)
a-p(i) where, p(i) is the ith prime below a. Thus, we have approximately n odd numbers below a.
We have n/a = 1/lna which gives the probability of prime number below a.
If we consider only the odd numbers below a, the probability becomes 2/lna, which means 2 out of lna odd numbers below a are prime numbers.
But we have generated approximately(which just excludes case for 2 and is negligible for large numbers) a/lna odd numbers.
So, from unitary method, 2/lna out of 1 odd numbers below a are prime, which yields for our case, $ 2 a/(lna)^2$ prime numbers out of a/lna odd numbers generated.
And we can see that, 2 a/(lna)^2 is obviously greater than 1 which even grows when a gets larger.
This shows that, among our generated odd numbers(a-p(i)) there is at least one prime number q giving,
a-p(i) = q
or, p(i) + q = a. where p(i) and q are primes, verifying Goldbach Conjecture.
Edit: I just wanted to know what is wrong with this approach(I knew there was). And I got my answer too. Thank you.