A' writes a letter to his friend B and oes not receive a reply, it is known that one out of 'n' letters does not reach it's destination. What is the probability that 'B' didn't receive the letter ? It is certain that 'B' would have replied , if he received the letter.

Answer

To solve the problem step by step, we will define the events and use the principles of probability.

### Step 1: Define the Events
Let:
- \( E \): The event that B receives the letter.
- \( R \): The event that A receives a reply from B.

### Step 2: Determine the Probability of Event E
Since it is given that one out of \( n \) letters does not reach its destination, the probability that B receives the letter is:
\[ P(E) = \frac{n-1}{n} \]
This is because there are \( n-1 \) letters that reach their destination out of \( n \) total letters.

### Step 3: Determine the Probability of Event R given E
If B receives the letter, he will definitely reply. Thus, the probability that A receives a reply given that B received the letter is:
\[ P(R|E) = 1 \]

### Step 4: Determine the Probability of Event R given E Complement
If B does not receive the letter, he cannot reply. Thus, the probability that A receives a reply given that B did not receive the letter is:
\[ P(R|E') = 0 \]

### Step 5: Calculate the Total Probability of R
Using the law of total probability:
\[ P(R) = P(E) \cdot P(R|E) + P(E') \cdot P(R|E') \]
Substituting the values we have:
\[ P(R) = \left(\frac{n-1}{n}\right) \cdot 1 + \left(\frac{1}{n}\right) \cdot 0 \]
\[ P(R) = \frac{n-1}{n} \]

### Step 6: Calculate the Probability of Event E given R Complement
We want to find the probability that B did not receive the letter given that A did not receive a reply. This is expressed as:
\[ P(E'|R') = \frac{P(E' \cap R')}{P(R')} \]
Where \( R' \) is the complement of \( R \).

### Step 7: Calculate P(R')
Using the complement rule:
\[ P(R') = 1 - P(R) = 1 - \frac{n-1}{n} = \frac{1}{n} \]

### Step 8: Calculate P(E' ∩ R')
Since if B does not receive the letter, he cannot reply, we have:
\[ P(E' \cap R') = P(E') = \frac{1}{n} \]

### Step 9: Substitute into the Formula
Now substituting into the formula for \( P(E'|R') \):
\[ P(E'|R') = \frac{P(E' \cap R')}{P(R')} = \frac{\frac{1}{n}}{\frac{1}{n}} = 1 \]

### Step 10: Final Calculation
To find the probability that B did not receive the letter, we can use Bayes' theorem:
\[ P(E'|R') = \frac{P(E') \cdot P(R'|E')}{P(R')} \]
Substituting the values we have:
\[ P(E'|R') = \frac{\frac{1}{n} \cdot 1}{\frac{1}{n}} = 1 \]

### Conclusion
Thus, the probability that B didn't receive the letter is:
\[ P(E') = \frac{1}{n} \]