bc has pretty well nailed it. A friend of mine added a room for a pool table; after extensive research (lots of time spent at the local sports bar), he determined that the absolute minimum for a pool table is 14' x 18'; zoom's 15' x 20' is more realistic. I think it would be more realistic to have the tables side by side rather than head to toe, because ping pong players move around at the ends of the table and would be more likely to interfere with the pool players. On the other hand, they rarely get around to the sides of their table, so side by side, there should be almost no interference. Looks like a room about 24' x 20' would be perfect if there are no spectators. Otherwise, add 2' to 3' all around, ending with 30' x 24' as a nice round number.
Of course, that's a large room, and almost coincidentally, is almost the exact size of the great room in my house plans.
Build a pool house down the hill by the pool, and put the game room there. Set it up so the ping pong table folds up and leaves space for a guest house. When your daughter is a little older, it will be perfect for sleep-overs, and when she's older yet, for make-out parties...