Question 1191532
Here's how to break down the probabilities:

**Scenario 1: B adds a positive message**

1. **A's message composition:** A starts with 6 positive (P) and 6 negative (N) messages.  A deletes one and adds a positive.  There are two possibilities:
    * A deletes a P:  A sends 7P, 5N
    * A deletes an N:  A sends 6P, 6N

2. **B's message composition:** B receives either 7P, 5N or 6P, 6N. B deletes one and adds a positive message. Again, there are a few possibilities:
    * B receives 7P, 5N:
        * B deletes a P: B sends 7P, 5N
        * B deletes an N: B sends 8P, 4N
    * B receives 6P, 6N:
        * B deletes a P: B sends 6P, 6N
        * B deletes an N: B sends 7P, 5N

3. **C's message composition:** C receives one of the message combinations from B. C deletes one and adds a positive message. We are interested in the probability that C has *fewer* negative messages than A.  Let's look at each of A's possible starting points and then B's possible actions to get to C:

    * **A sends 7P, 5N:**
        * B deletes a P (7P, 5N): C can get 7P, 5N or 8P, 4N, which are the same or less negative than A.
        * B deletes an N (8P, 4N): C can get 8P, 4N or 9P, 3N, which are the same or less negative than A.
    * **A sends 6P, 6N:**
        * B deletes a P (6P, 6N): C can get 6P, 6N or 7P, 5N, which are the same or less negative than A.
        * B deletes an N (7P, 5N): C can get 7P, 5N or 8P, 4N, which are the same or less negative than A.

In *every single case*, C ends up with the same or *fewer* negative messages than A.  Therefore, the probability of C having fewer negative messages than A is 1 (or 100%).

**Scenario 2: B adds a negative message**

1. **A's message composition:** Same as before: 7P, 5N (if A deletes a P) or 6P, 6N (if A deletes an N).

2. **B's message composition:** B receives 7P, 5N or 6P, 6N and deletes one message and adds a negative.
    * B receives 7P, 5N:
        * B deletes a P: 7P, 6N
        * B deletes an N: 6P, 7N
    * B receives 6P, 6N:
        * B deletes a P: 6P, 7N
        * B deletes an N: 5P, 8N

3. **C's message composition:**  C receives one of the above combinations, deletes one, and adds a positive. Let's examine each of A's starting points:

    * **A sends 7P, 5N:**
        * B sends 7P, 6N: C can get 7P, 6N or 8P, 5N. Both have the same or more negative messages than A.
        * B sends 6P, 7N: C can get 6P, 7N or 7P, 6N. Both have the same or more negative messages than A.
    * **A sends 6P, 6N:**
        * B sends 6P, 7N: C can get 6P, 7N or 7P, 6N. Both have the same or more negative messages than A.
        * B sends 5P, 8N: C can get 5P, 8N or 6P, 7N. Both have the same or more negative messages than A.

In *every single case*, C ends up with the same or *more* negative messages than A.  Therefore, the probability of C having *fewer* negative messages than A is 0.