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 time it takes to finish a 675-word passage, we need to calculate the time it takes to type the correct words and the time it takes to correct the mistakes.
First, let's calculate the time it takes to type the correct words. Since the typist types at a constant speed of 65 words per minute, we can calculate the time it takes to type 675 words using the formula:
Time to type correct words = Number of words / Typing speed
Time to type correct words = 675 words / 65 words per minute
Time to type correct words ≈ 10.38 minutes
But we need the time in seconds, so we multiply by 60:
Time to type correct words = 10.38 minutes * 60 seconds per minute
Time to type correct words ≈ 622.8 seconds
Next, let's calculate the time it takes to correct the mistakes. Since each incorrect word needs to be corrected, the time it takes to correct the mistakes can be calculated as:
Time to correct mistakes = Number of mistakes * Time to correct each mistake
To find the number of mistakes, we multiply the number of correct words by the probability of making a mistake:
Number of mistakes = Number of words * Probability of mistake
Number of mistakes = 675 words * 0.06
Number of mistakes = 40.5 (rounded to the nearest whole number)
Time to correct mistakes = 40 mistakes * 14 seconds per mistake
Time to correct mistakes = 560 seconds
Finally, we can find the mean time it takes to finish the passage by adding the time to type the correct words and the time to correct the mistakes:
Mean time = Time to type correct words + Time to correct mistakes
Mean time ≈ 622.8 seconds + 560 seconds
Mean time ≈ 1182.8 seconds
Rounding off to the nearest whole number, the mean time it takes to finish a 675-word passage is approximately 1183 seconds.
To find the mean of the time it takes to finish a 675-word passage, we need to consider the time it takes to type and correct each word.
Let's break down the steps involved:
1. Calculate the average time to type a word:
The typist types at a constant speed of 65 words per minute. Therefore, the average time to type a word is 60 seconds divided by 65 words, which is approximately 0.9231 seconds per word.
2. Calculate the average time to correct a word:
The typist makes a mistake with probability 0.06. This means that for each word, there is a 0.06 probability that it needs to be corrected. Since each incorrect word takes 14 seconds to correct, the average time to correct a word is 0.06 * 14 = 0.84 seconds per word.
3. Calculate the total time for typing and correcting the entire passage:
Since each word requires both typing and, if necessary, correcting, the total time per word is the sum of the average typing time and the average correction time, which is 0.9231 + 0.84 = 1.7631 seconds per word.
Therefore, the total time to type and correct a 675-word passage is 1.7631 * 675 = 1188.6535 seconds.
Rounding off to the nearest integer, the mean of the time it takes to finish the passage is approximately 1189 seconds.