Question 1192025
Here's how to break down this probability problem:

**a) B deletes a message (any message) and adds a good one:**

* **A's messages:** 6 good, 6 bad (12 total)
* **B's messages:**  B receives 12, deletes 1, and adds 1 good.  There are three scenarios for the deleted message from A:
    * **Scenario 1 (Deleted Good):** B receives 5 good, 6 bad.  Adds 1 good = 6 good, 6 bad.
    * **Scenario 2 (Deleted Bad):** B receives 6 good, 5 bad. Adds 1 good = 7 good, 5 bad.
* **C's messages:** C receives 12, deletes 1, and adds 1 good. Again, two scenarios for the deleted message from B:
    * **Scenario 1.1 (Deleted Good from B - Scenario 1):** C receives 5 good, 6 bad. Adds 1 good = 6 good, 6 bad.
    * **Scenario 1.2 (Deleted Bad from B - Scenario 1):** C receives 6 good, 5 bad. Adds 1 good = 7 good, 5 bad.
    * **Scenario 2.1 (Deleted Good from B - Scenario 2):** C receives 6 good, 5 bad. Adds 1 good = 7 good, 5 bad.
    * **Scenario 2.2 (Deleted Bad from B - Scenario 2):** C receives 7 good, 4 bad. Adds 1 good = 8 good, 4 bad.

* **Probability Calculation:** Each of A's deleted message scenarios has a 1/2 probability. From there, each of B's deleted message scenarios has a 1/2 probability. So each final outcome has a 1/4 probability.
    * C has 6 bad messages (Scenario 1.1): Probability = 1/4
    * C has 5 bad messages (Scenarios 1.2 & 2.1): Probability = 1/4 + 1/4 = 1/2
    * C has 4 bad messages (Scenario 2.2): Probability = 1/4

* **C has *less* bad messages than A (6):** This includes the cases where C has 5 or 4 bad messages.
* **Probability (C < A) = 1/2 + 1/4 = 3/4 = 0.75**

**b) B deletes a bad message and adds a bad one:**

* **A's messages:** 6 good, 6 bad
* **B's messages:** B receives 12, deletes 1 bad, adds 1 bad. B has 6 good, 6 bad messages.
* **C's messages:** C receives 12, deletes 1. Two scenarios:
    * **Scenario 1 (Deleted Good from B):** C receives 5 good, 6 bad. Adds 1 good = 6 good, 6 bad.
    * **Scenario 2 (Deleted Bad from B):** C receives 6 good, 5 bad. Adds 1 good = 7 good, 5 bad.

* **Probability Calculation:** Each of the scenarios for C has a 1/2 probability.
    * C has 6 bad messages (Scenario 1): Probability = 1/2
    * C has 5 bad messages (Scenario 2): Probability = 1/2

* **C has *less* bad messages than A (6):**  This only occurs when C has 5 bad messages.
* **Probability (C < A) = 1/2 = 0.50**