A game is played with a pair of unique six-faced dice. The dice are independent and fair, but are numbered 1, 1, 2, 2, 3, 3 instead of 1 to 6. The game begins with a payment of $4. Then, the two dice are tossed simultaneously. The numbers on the top faces of the dice are observed. The player is awarded Z, where Z equals the sum of the two numbers. Calculate the expected profit from one round of this game.