I think your calculation of the odds of a number which reads the same backwards as forwards is wrong, but your first point is quite powerful. (The odds of a number which reads forward and backwards being divisible by a number would be the number of such numbers which are divisible divided by the number which have a common divisor. If you exclude primes and only focus on large numbers, this ratio is going to be large as it includes all numbers with a large factor in common, as that common factor is likely to have a common divisor less than 10 as that's just the way factoring numbers works. So.... I guess it comes down to [total of all large forward-backward pairs] - [total number that don't have a common divisor(?)] / [total of all large forward-backward pairs]. I think..... )
(?) Is that the same as relatively prime?