Question 1192025: A received 6 good and 6 bad messages, accidentally deleted one, added one good and sent to B. B also accidentally deleted one message and sent one good message to C.
a) what is the probability that C will receive less bad messages than A?
b) what is the probability that C will receive fewer bad messages than A if B accidentally deleted one and added one bad message?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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**
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