There are 4 positions in the string that each have 25 choices (any letter in the English alphabet except A) and the other positions are filled with A, so there are 25⁴ possible strings containing exactly 7 A's.
Overall, there are 26¹¹ possible strings of length 11.
Then the probability of randomly generating a string containing exactly 7 A's is
25⁴ / 26¹¹ ≈ 0.000000000106427
