expected value of a random palindrome If you choose a 6-digit palindrome at random, what is the expected value for the number? 
All possible palindromes are equally likely to be chosen. Beginning number must be NONZERO, so numbers like 012321 are NOT allowed.
I'm not sure where to start. What values and the probabilities of the values should I look at? 

Well, first of all, you need to count all possible outcomes.
In other words, you need to count all possible palindromes. All your palindromes are uniquely defined by their first three digits. You have $9$ possibilities for the first digit (can't be zero) and $10$ possibilities for the second and third. 
Secondly, since the numbers are chosen with equal probabilities, you need to count their sum. The hint for this part is that $${abccba} = 100001\cdot a + 10010\cdot b + 1100\cdot c.$$
Can you take it from here?
