2010 AMC 12B Problem 11

A four-digit palindrome has the form $\overline{abba}=1001a+110b$ with $1\le a\le9$ and $0\le b\le9.$ Since $1001=7\cdot11\cdot13$ is divisible by $7$ and $110$ is not, the number is divisible by $7$ exactly when $b$ is divisible by $7,$ that is $b=0$ or $b=7.$ For each $a,$ that is $2$ of the $10$ choices of $b,$ a probability of $\dfrac{2}{10}=\dfrac15.$ Thus, the correct answer is E.