Calculate the expression: -2 + (11/24 + 5/6) : (15/16 : 5/2)

1. Calculate the sum inside the first parenthesis: 11/24 + 5/6 = 11/24 + 20/24 = 31/24.

2. Calculate the division inside the second parenthesis: 15/16 : 5/2 = 15/16 * 2/5 = 30/80 = 3/8.

3. Perform the division: 31/24 : 3/8 = 31/24 * 8/3 = 248/72 = 31/9.

4. Perform the final addition: -2 + 31/9 = -18/9 + 31/9 = 13/9.

Final Answer: The final answer is $$\boxed{\frac{13}{9}}$$