Question

7. According to Maryland Motor Vehicle Administration [MVA] data, Gary Turgeon, a clerk at the Beltsville, Maryland, MVA location, assists three customers per hour, on average. a. Determine the probability the amount of time Gary takes to assist the next customer is between 6 and 12 minutes (in the interval 6 to 12 minutes). b. Determine the probability the amount of time Gary takes to assist the next customer is between 26 and 35 minutes (in the interval 26 to 35 minutes). c. Determine the probability the amount of time Gary takes to assist the next customer is either less than 14 minutes or greater than 24 minutes.

Answer 1

The probability that Gary assists a customer in 6 to 12 minutes is 30%, between 26 to 35 minutes is 0%, and either less than 14 minutes or greater than 24 minutes is 70%.

Step-by-step explanation:

Probability of Service Times

Given that Gary Turgeon assists three customers per hour, we deduce that he assists one customer every 20 minutes on average. This is because there are 60 minutes in an hour, so to assist three customers, it would take 60 / 3 = 20 minutes per customer.

a. Probability between 6 and 12 minutes

Assuming a uniform distribution of service times, the probability that Gary takes between 6 and 12 minutes is the proportion of the interval 6 to 12 minutes within the average service time:

P(6 < X < 12) = (12 - 6) / (20 - 0) = 6 / 20 = 0.30 or 30%

b. Probability between 26 and 35 minutes

This interval lies entirely outside the average service time of 20 minutes. Therefore, the probability is:

P(26 < X < 35) = 0 or 0%

c. Probability less than 14 minutes or greater than 24 minutes

We sum the probabilities of both intervals:

P(X < 14) = 14 / 20 = 0.70 or 70%

P(X > 24) is theoretically 0, since it's beyond the average service interval. But if we are considering the possibility of service times longer than average:

P(X > 24) + P(X < 24) = 1
P(X > 24) = 1 - (24 / 20)
Since 24 / 20 exceeds 1, this calculation is not meaningful in the context of a uniform distribution with a maximum time of 20 minutes, so we regard the probability as 0.

Combining the two probabilities:

P(X < 14 or X > 24) = P(X < 14) + P(X > 24) = 0.70 + 0 = 0.70 or 70%

Answer 2

The probability that the amount of time Gary takes to assist the next customer is between 6 and 12 minutes is 3. The probability that the amount of time Gary takes to assist the next customer is between 26 and 35 minutes is 4.5. The probability that the amount of time Gary takes to assist the next customer is either less than 14 minutes or greater than 24 minutes is 0.

Step-by-step explanation:

To determine the probability that the amount of time Gary takes to assist the next customer is between 6 and 12 minutes, we need to calculate the cumulative probability of 12 minutes and subtract the cumulative probability of 6 minutes. Using the given information, we can calculate the cumulative probability of 6 minutes as follows:

Cumulative probability = (6 - 2)/2 = 4/2 = 2

And the cumulative probability of 12 minutes:

Cumulative probability = (12 - 2)/2 = 10/2 = 5

Subtracting the cumulative probability of 6 minutes from the cumulative probability of 12 minutes gives us:

Probability = 5 - 2 = 3

Therefore, the probability that the amount of time Gary takes to assist the next customer is between 6 and 12 minutes is 3.

To determine the probability that the amount of time Gary takes to assist the next customer is between 26 and 35 minutes, we can use the same approach. Calculate the cumulative probability of 26 minutes and subtract the cumulative probability of 35 minutes:

Cumulative probability of 26 minutes = (26 - 2)/2 = 24/2 = 12

Cumulative probability of 35 minutes = (35 - 2)/2 = 33/2 = 16.5

Subtracting the cumulative probability of 26 minutes from the cumulative probability of 35 minutes:

Probability = 16.5 - 12 = 4.5

Therefore, the probability that the amount of time Gary takes to assist the next customer is between 26 and 35 minutes is 4.5.

To determine the probability that the amount of time Gary takes to assist the next customer is either less than 14 minutes or greater than 24 minutes, we can calculate the cumulative probability of 14 minutes and subtract it from 1 to get the probability of the time being greater than 14 minutes.

Then we can calculate the cumulative probability of 24 minutes and subtract it from 1 to get the probability of the time being greater than 24 minutes. Finally, we can add these probabilities together to get the overall probability:

Probability of time less than 14 minutes = (14 - 2)/2 = 12/2 = 6

Probability of time greater than 14 minutes = 1 - 6 = 0

Probability of time less than 24 minutes = (24 - 2)/2 = 22/2 = 11

Probability of time greater than 24 minutes = 1 - 11 = 0

The overall probability is 0 + 0 = 0. Therefore, the probability that the amount of time Gary takes to assist the next customer is either less than 14 minutes or greater than 24 minutes is 0.