A typist at a constant speed of 65 words per minute makes a mistake in any particular word with probability 0.06, independently from word to word. Each incorrect word must be corrected, a task which takes 14 seconds per word. Find the mean of the time (in seconds, round-off to an integer) it takes to finish a 675-word passage.

To find the mean of the time it takes to finish a 675-word passage, we need to calculate the total time it takes for each word, including the time taken to correct any mistakes.

Let's break down the different components of time involved:

1. Time to type a word without a mistake:

Since the typist types at a constant speed of 65 words per minute, the time taken to type one word correctly is:
Time to type one word correctly = (1 word) / (65 words/minute) = (1/65) minutes

To convert this into seconds, we multiply by 60:
Time to type one word correctly = (1/65) * 60 seconds = (60/65) seconds

2. Time to type a word with a mistake:

Since the probability of making a mistake in any particular word is 0.06, the probability of typing a word correctly is 1 - 0.06 = 0.94.

Therefore, the time taken to type a word with a mistake is the sum of the time taken to type a word correctly and the time taken to correct the mistake:
Time to type one word with a mistake = Time to type one word correctly + Time taken to correct the mistake
Time to type one word with a mistake = (60/65) seconds + 14 seconds

3. Total time to finish a 675-word passage:

In a 675-word passage, the typist can either type a word correctly or type a word with a mistake. The probability of typing a word correctly is 0.94, and the probability of typing a word with a mistake is 0.06.

Therefore, the total time to finish a 675-word passage is calculated as follows:

Total time = (Time to type one word correctly * Number of correct words) + (Time to type one word with a mistake * Number of words with mistakes)
Total time = ((60/65) * 0.94 * 675 seconds) + ((60/65 + 14) * 0.06 * 675 seconds)

Calculating this expression will give us the mean of the time it takes to finish a 675-word passage.