Question 1191532: Person A has recieved 6 negative and 6 positive text messages. Person A randomly deletes one of the messages and adds one positive one to the bunch and sends all of the messages to person B. B also deletes one random message and adds a positive one to the bunch and sends them all to person C. What is the probability of C receiving less negative messages than A. What is the probability of C getting less negative messages than A if B instead of adding a positive message, adds a negative one?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
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