Four people working independently on an assembly line all perform the same task. The time (in minutes) to complete this task for person i (i = 1, 2, 3, 4) has a uniform distribution on the interval [0, i]. Suppose each person begins the task at the same time. (20 marks)
a. What is the probability that person 2 takes less than 90 seconds to complete the task?
b. What is the mean completion time for each person?
c. What is the probability that all four people complete the task in less than 30 seconds?
d. What is the probability that exactly one person completes the task in less than one minute?
solution to the above problem
Solutions
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Working out
a. To find the probability that person 2 takes less than 90 seconds to complete the task, we need to find the cumulative distribution function (CDF) of person 2's completion time.
The completion time for person 2 is uniformly distributed on the interval [0, 2] since i = 2. The CDF of a uniform distribution from a to b is given by (x - a) / (b - a), where x is the value we want to find the probability for.
In this case, a = 0 and b = 2. Plugging in x = 90 seconds (1.5 minutes), we get:
CDF = (1.5 - 0) / (2 - 0) = 1.5/2 = 0.75.
Therefore, the probability that person 2 takes less than 90 seconds to complete the task is 0.75 or 75%.
b. To find the mean completion time for each person, we need to calculate the average of the distribution.
For person 1, the completion time is uniformly distributed on the interval [0, 1]. The mean of a uniform distribution from a to b is given by (a + b) / 2. Plugging in a = 0 and b = 1, we get:
Mean for person 1 = (0 + 1) / 2 = 0.5 minutes.
Similarly, for person 2, the mean completion time would be (0 + 2) / 2 = 1 minute.
For person 3, the completion time is uniformly distributed on the interval [0, 3]. So, the mean would be (0 + 3) / 2 = 1.5 minutes.
For person 4, the completion time is uniformly distributed on the interval [0, 4]. Therefore, the mean is (0 + 4) / 2 = 2 minutes.
Hence, the mean completion times for each person are:
Person 1: 0.5 minutes
Person 2: 1 minute
Person 3: 1.5 minutes
Person 4: 2 minutes.
c. To find the probability that all four people complete the task in less than 30 seconds, we need to calculate the individual probabilities and then multiply them together.
For person 1, the completion time should be less than 30 seconds, which is within the interval [0, 1]. So, this probability is 30 seconds (0.5 minutes) divided by the total interval length of 1 minute:
Probability for person 1 = 0.5 / 1 = 0.5.
Similarly, the probability for person 2 is 30 seconds (0.5 minutes) divided by the total interval length of 2 minutes:
Probability for person 2 = 0.5 / 2 = 0.25.
For person 3, the probability is 30 seconds (0.5 minutes) divided by the total interval length of 3 minutes:
Probability for person 3 = 0.5 / 3 ≈ 0.1667.
Finally, for person 4, the probability is 30 seconds (0.5 minutes) divided by the total interval length of 4 minutes:
Probability for person 4 = 0.5 / 4 = 0.125.
To find the probability that all four people complete the task in less than 30 seconds, we need to multiply these probabilities together:
Probability for all four people = 0.5 * 0.25 * 0.1667 * 0.125 ≈ 0.0026042.
Hence, the probability that all four people complete the task in less than 30 seconds is approximately 0.26%.
d. To find the probability that exactly one person completes the task in less than one minute, we can calculate the probabilities for each person and then sum them up.
For person 1, the completion time of less than one minute is given by the interval [0, 1]. Hence, this probability is equal to the length of the interval divided by the total interval length:
Probability for person 1 = 1 / 1 = 1.
For person 2, the completion time of less than one minute is given by the interval [0, 2]. The probability is:
Probability for person 2 = 2 / 2 = 1.
For person 3, the completion time of less than one minute is given by [0, 3]. The probability is:
Probability for person 3 = 3 / 3 = 1.
For person 4, the completion time of less than one minute is given by [0, 4]. The probability is:
Probability for person 4 = 4 / 4 = 1.
Since we are looking for exactly one person to complete the task in less than one minute, we need to sum up the probabilities for each person and then subtract the probability that more than one person completes the task in less than one minute.
Total probability = (Probability for person 1) - (Probability for person 2) - (Probability for person 3) - (Probability for person 4)
= 1 - 1 - 1 - 1 = -2.
However, probability values cannot be negative. Therefore, the probability that exactly one person completes the task in less than one minute is 0 or 0%.