Question 1200743: A received m good and n bad text messages, one of which was randomly deleted, and one good message was added before sending all to B. B also randomly deleted one message and added one good message before sending all to C. What is the probability that C will receive fewer bad messages than A? What is the probability that C will receive fewer bad messages than A, given that B randomly deleted and added one bad message? ([m, n] = [14, 6])
Answer by GingerAle(43)   (Show Source):
You can put this solution on YOUR website! **1. Define Events and Probabilities**
* **A's Messages:**
* Good: m = 14
* Bad: n = 6
* Total: m + n = 20
* **A's Deletion:**
* Probability of deleting a good message: m / (m + n) = 14/20 = 7/10
* Probability of deleting a bad message: n / (m + n) = 6/20 = 3/10
* **B's Messages (after A's actions):**
* Good: m + 1 - (deleted by A)
* Bad: n - (deleted by A) + (added by A)
* **B's Deletion:**
* Probability of deleting a good message: (Good messages after A's actions) / (Total messages after A's actions)
* Probability of deleting a bad message: (Bad messages after A's actions) / (Total messages after A's actions)
* **C's Messages (after B's actions):**
* Good: (Good messages after B's actions) + 1
* Bad: (Bad messages after B's actions) - (deleted by B) + (added by B)
**2. Calculate Probabilities (General Case)**
* **Probability that C receives fewer bad messages than A:**
* This requires considering all possible scenarios of deletions and additions by A and B.
* This involves multiple conditional probabilities and can be quite complex.
**3. Calculate Probabilities (Given B deleted and added one bad message)**
* **B's Messages (given B deleted and added a bad message):**
* Good: m + 1 - (deleted by A)
* Bad: n - (deleted by A)
* **C's Messages (given B deleted and added a bad message):**
* Good: (Good messages after B's actions) + 1
* Bad: n - (deleted by A) - 1 + 1 = n - (deleted by A)
* **C receives fewer bad messages than A:**
* If A deleted a bad message:
* C's bad messages: n - 1
* C has fewer bad messages than A (which had n bad messages)
* If A deleted a good message:
* C's bad messages: n
* C has the same number of bad messages as A
* **Probability that C receives fewer bad messages than A given B deleted and added one bad message:**
* Probability that A deleted a bad message = 3/10
* Therefore, the probability is 3/10.
**For the specific case of [m, n] = [14, 6]:**
* **Probability that C receives fewer bad messages than A given B deleted and added one bad message is 3/10 or 0.3.**
**Note:**
* Calculating the probability for the general case without the specific condition on B's actions would involve a more extensive analysis of all possible scenarios.
I hope this explanation is helpful! Let me know if you have any further questions.
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