I think your answer is definitely correct (I got the same answer doing it a different way than you).

I think it's your checking that's gone wrong! I don't really understand where you're getting your numbers in the second part. The probabilities are

A) Miss first and second (40/47)*(36/46)=1440/2162

B) Miss first, hit second (40/47)*(10/46)=400/2162

C) Hit first (no matter what happens with second) 7/47 = 322/2162

These add up just fine.

Edit because I think I was wrong:

If you take 10/46+7/47-(7/47*10/46) = 722/2162 you will notice that this is the same as 1-P(A) (what you wanted).