This is a 0-1 box. The probability of a 1 is 3/4. Therefore its name is B(1, 3/4).
The possible outcomes are 0, 1, and 2.
F # F is a box containing pairs of results, one from F and the second from F. Therefore it contains 4 * 4 = 16 tickets. One of them has (0, 0). Three of them have (1, 0) and three of them have (0, 1). Nine of them have (1, 1).
Adding changes (0, 0) to 0+0 = 0, (0, 1) to 0+1 = 1, (1, 0) to 1+0 = 1, and (1, 1) to 1+1 = 2.
Therefore there is one ticket with 0, 3+3 tickets with 1, and nine tickets with 2. Their probabilities are 1/16, 6/16, and 9/16.
The name of the sum box is B(2, 3/4). In general, the sum of a B(1, p) box and another B(1, p) box is called a B(2, p) box. If p is not exactly 0 or 1, its outcomes are 0, 1, and 2. Their probabilities are (1-p)2, 2*p*(1-p), and p2, respectively. You can verify this formula for the case of B(2, 3/4) above.