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 expected time for each word and then sum them up.
First, let's calculate the expected time for each word.
The typist types at a constant speed of 65 words per minute, so the time to type one word is 1/65 minutes or 60/65 seconds.
The probability of making a mistake in any particular word is 0.06. So, the probability of typing a word correctly is 1 - 0.06 = 0.94.
If the typist types a word correctly, no correction is needed, so the time taken for that word is 60/65 seconds.
If the typist makes a mistake and needs to correct the word, it takes an additional 14 seconds. Since the probability of making a mistake is 0.06, the expected time for an incorrect word is 14 seconds * 0.06 = 0.84 seconds.
The expected time for each word can be calculated as follows:
Expected Time = (Probability of typing correctly * Time for correct word) + (Probability of making a mistake * Time for incorrect word)
Expected Time = (0.94 * 60/65 seconds) + (0.06 * 14 seconds)
Expected Time = (0.94 * 60/65) + (0.06 * 14)
Expected Time ≈ 54.4615 seconds
Now, let's calculate the mean time to finish the 675-word passage.
Mean Time = Number of words * Expected Time
Mean Time = 675 * 54.4615
Mean Time ≈ 36738 seconds
Rounded off to an integer, the mean time it takes to finish the 675-word passage is approximately 36738 seconds.