In this analysis, I start with a few assumptions :
From this I want to show that we are the first star to evolve intelligence.
For that I need one additional assumption :
Here is the "proof".
If we put the moment of formation of the first star at 0MY, we are now at a moment in time we might call 7000MY. Since no intelligence has visited us, the first possible intelligence must have evolved less than 1MY ago. Let's assume that indeed the first intelligence, out of 1,000,000, evolved at 6999MY.
It is impossible to determine the lowest of a set of normally distributed events, but we can calculate the point at which the probability that there is any event smaller, is equal to 1 millionth. In a normal distribution, this happens at a point 4.75 times the standard deviation away from the average.
So if the standard deviation of our normal distribution is termed S, the average value of our distribution M will be 6999 + 4.75S.
Now when would the second intelligence emerge in that time scale ?
Well, the point at which the probability becomes 2 in a million, is 4.61 times the standard deviation.
So one would normally expect the second intelligence to evolve at M - 4.61S or 6999+0.14S.
If we now suggest that this second intelligence has in fact come at 7000MY, we find that S would equal about 7MY.
That would mean that the average value of our distribution would be at 7033MY, and that 95% of all intelligences would arrive between 7019MY and 7047MY.
I find that hard to believe, don't you ?
If you don't, consider that some stars began their evolution at 0MY, others, like our sun, around 2000MY, still others probably much later. What mechanism could possibly force the evolutions on younger planets to be faster than ours ? (or that of older stars to be slower ?)
I realise that this is not a rigourously mathematical proof, of course. I hope that you accept it nevertheless, and perhaps someone with more correct knowledge can turn this into a calculation of likelyhood, so that I can turn it into : "the probability that the first intelligence was formed at 6999, given that a second is formed at 7000, is -such and such small number- if we assume the standard deviation to be (some believable number)".
Please realise too that a similar "proof" shows that it is extremely unlikely that we (if we are the first) will find a second intelligence in the next 1MY !
Back to the main argument.