Challenge from Westdene Primary, Brighton

Age 7 to 14 Challenge Level:
Tom Kimber, a teacher at Westdene Primary School in Brighton emailed us to say:

I am a MaST teacher who was lucky enough to be at one of your recent NRICH events in London. I got into a really interesting mathematical discussion with some very advanced pupils in my class about the pattern of the units column in numbers-to-the-power-of.

Anyway, after a long and convoluted journey through many different mathematical areas, it has reached a concluding* question. I thought I'd share it with you as my children have ENJOYED getting their teeth into it:

Look at this calculation:

$7^x + 9^y =$

I get a random number generator to generate two random whole numbers. The first one it generates I place in position $x$. The second number it generates I put at position $y$. What is the probability that, when I complete the calculation $7^x + 9^y =$, I will get a number that is a multiple of $10$ as my answer?

Hope you find it as interesting as we are!

* - I say concluding... probably not!