What are the probabilities for more than +/- 3 standard deviations away from the mean in a normal distribution? With a normal distribution, I'm familiar with +/- 1, 2, and 3 standard deviations:
+/-
1: 68%
2: 95.45%
3: 99.73%
What are the probabilities for +/- 4, 5, 6, 7, 8, 9, or even 10 standard deviations away from the mean?

We can figure this out by using the cumulative density function of the normal distribution.
In math notation, $\Psi(n)-\Psi(-n)$, where $\Psi$ is the cdf and $n$ the number of standard deviations.
In R for example:
(pnorm(1:10)-pnorm(-(1:10)))

0.6826895 0.9544997 0.9973002 0.9999367 0.9999994 1.0000000 1.0000000 1.0000000 1.0000000
As we can see, it goes to 1 very quickly (though never quite reaches it; we observe exactly 1 due to numerical error).
