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Assume that in the group of integers $\mathbb{Z}$, I randomly choose two integers $a$ and $b$ and I would like to ask whether they generate the group $\mathbb{Z}$, and then, what is the probability for this event? However, to explain clearly "probability meaning", we need to know about probability distribution. Could any one give me an exact definition for this then? Or here, we just reduce our consideration on each quotient group $\mathbb{Z}/n\mathbb{Z}$ for each $n$?

Thanks in advance.

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You could ask, for each integer $N$, what is the probability that two integers chose uniformly at random between $0$ and $N$ generate the group, then you could ask for the limit as $N\to\infty$. But they generate if and only if they are relatively prime, and the probability of this is known to be $\pi^2/6$. – Gerry Myerson Nov 30 '12 at 11:43
@Gerry: $\pi^2/6 \gt 1$ – Henry Nov 30 '12 at 11:46
Sorry, $6/\pi^2$. – Gerry Myerson Nov 30 '12 at 21:44
up vote 9 down vote accepted

There is no uniform distribution on the integers.

But if you take two random integers uniformly from the interval $[-n,n]$, then the limit of the probability that they are coprime is $\dfrac{1}{\zeta(2)}=\dfrac{6}{\pi^2} \approx 0.6079$.

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Thank Henry, that's what I am thinking of. – 9999 Nov 30 '12 at 14:20

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