jmc

The best way to do this is to find the probability that the two E's are next to each other. There are $\frac{7!}{2}$ arrangements of the word SIXTEEN. If we want to find the number of arrangements such that the E's are next to each other, we find the number of arrangements of the six-letter word SIXT(EE)N (where we treat the two E's as a block), which is $6!$. So the probability that an arrangement of the word SIXTEEN has the two E's next to each other is $\frac{6!}{\frac{7!}{2}}=\frac{2}{7}$. So the probability that the two E's aren't next to each other is $1-\frac{2}{7}=\boxed{\frac{5}{7}}$.