Mr. A writes an article. The article originally is error free. Each day Mr. B introduces one new error into the article. At the end of the day, Mr. A checks the article and has `(2)/(3)` chance of catching each individual error still in the article. After 3 days, the probability that the article is error free can be expressed as `(p)/(q)` where p and q are relatively prime positive integers. Let `lambda=q-p`, then find the sum of the digits of `lambda`.

Answer

To solve the problem step by step, we need to calculate the probability that the article is error-free after 3 days, given the conditions provided.

### Step 1: Understand the Problem
- Mr. A writes an article that is initially error-free.
- Each day, Mr. B introduces one new error into the article.
- Mr. A has a \( \frac{2}{3} \) chance of catching each error still in the article at the end of the day.

### Step 2: Determine the Errors Introduced
After 3 days, a total of 3 errors will have been introduced into the article (one each day).

### Step 3: Calculate the Probability of Catching Errors
For each error introduced, there are two scenarios:
1. The error is caught by Mr. A.
2. The error is not caught by Mr. A.

The probability of catching an error is \( \frac{2}{3} \), and the probability of not catching an error is \( \frac{1}{3} \).

### Step 4: Calculate the Probability of the Article Being Error-Free
To be error-free after 3 days, Mr. A must catch all 3 errors. We can calculate the probability of catching each error as follows:

- The probability that all 3 errors are caught:
\[ P(\text{error-free}) = P(\text{catch error 1}) \times P(\text{catch error 2}) \times P(\text{catch error 3}) = \left(\frac{2}{3}\right)^3 = \frac{8}{27} \]

- The probability that at least one error is not caught (i.e., the article is not error-free) can be calculated using the complement:
\[ P(\text{not error-free}) = 1 - P(\text{error-free}) = 1 - \frac{8}{27} = \frac{19}{27} \]

### Step 5: Calculate the Probability of Catching Errors
Now, we need to consider all possible combinations of catching the errors over the 3 days. The article can be error-free if:
1. All errors are caught.
2. At least one error is not caught.

The probability of the article being error-free after 3 days can be calculated by considering the different scenarios of catching errors.

### Step 6: Calculate the Total Probability
The total probability of the article being error-free after 3 days is given by:
\[ P(\text{error-free}) = \left(\frac{2}{3}\right)^3 = \frac{8}{27} \]

### Step 7: Express the Probability in Simplified Form
The probability \( \frac{8}{27} \) is already in its simplest form where \( p = 8 \) and \( q = 27 \).

### Step 8: Calculate \( \lambda \)
We need to find \( \lambda = q - p \):
\[ \lambda = 27 - 8 = 19 \]

### Step 9: Find the Sum of the Digits of \( \lambda \)
The sum of the digits of \( \lambda = 19 \) is:
\[ 1 + 9 = 10 \]

### Final Answer
The sum of the digits of \( \lambda \) is \( 10 \).