The arrival time of an elevator in a 12-story dormitory is equally likely at any time during the next 4.4 minutes.
a. Calculate the expected arrival time. (Round your answer to 2 decimal places.)
b. What is the probability that an elevator arrives in less than 1.1 minutes? (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)
c. What is the probability that the wait for an elevator is more than 1.1 minutes? (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)
a. The expected arrival time can be calculated by taking the average of the minimum and maximum possible arrival times. Since the arrival time is equally likely at any time during the next 4.4 minutes, the minimum possible arrival time is 0 minutes and the maximum possible arrival time is 4.4 minutes.
Expected arrival time = (0 + 4.4) / 2 = 2.2 minutes
b. The probability that an elevator arrives in less than 1.1 minutes can be calculated by dividing the desired time range (less than 1.1 minutes) by the total time range (4.4 minutes).
Probability = (1.1 / 4.4) = 0.25
c. The probability that the wait for an elevator is more than 1.1 minutes can be calculated by subtracting the probability of the elevator arriving in less than 1.1 minutes from 1.
Probability = 1 - 0.25 = 0.75
a. To calculate the expected arrival time, we need to find the average time the elevator takes to arrive. Since the arrival time is equally likely at any point during the next 4.4 minutes, the expected arrival time is simply the midpoint of the interval, which is 4.4/2 = 2.2 minutes.
b. To calculate the probability that the elevator arrives in less than 1.1 minutes, we need to find the proportion of the total time interval that is less than 1.1 minutes. To do this, we divide the desired time period (1.1 minutes) by the total time interval (4.4 minutes):
P(arrival in less than 1.1 minutes) = 1.1 / 4.4 = 0.25 (rounded to 2 decimal places).
c. To calculate the probability that the wait for an elevator is more than 1.1 minutes, we need to find the proportion of the total time interval that is greater than 1.1 minutes. To do this, we subtract the probability from part b from 1:
P(wait for an elevator is more than 1.1 minutes) = 1 - 0.25 = 0.75 (rounded to 2 decimal places).
a. To calculate the expected arrival time, we can use the formula:
Expected Value = (Sum of all possible outcomes * Probability of each outcome)
In this case, the possible outcomes are the arrival times of the elevator, which can be any time during the next 4.4 minutes. Since the arrival time is equally likely at any time, we can assume it follows a uniform distribution.
The uniform distribution has a constant probability density function, so we can calculate the expected value by taking the average of the minimum and maximum possible values.
The minimum possible arrival time is 0 minutes since the elevator can arrive immediately, and the maximum possible arrival time is 4.4 minutes.
Expected Value = (0 + 4.4) / 2 = 2.2
Therefore, the expected arrival time is 2.2 minutes.
b. To calculate the probability that an elevator arrives in less than 1.1 minutes, we need to determine the area under the probability density function for the given range.
Since the arrival time is equally likely at any time during the next 4.4 minutes and follows a uniform distribution, the probability density function is a rectangle with a base equal to the range of possible arrival times, which is 4.4 minutes in this case.
To calculate the probability, we divide the length of the desired range (1.1 minutes) by the total length of the distribution (4.4 minutes).
Probability = (Length of desired range) / (Total length of distribution)
Probability = 1.1 / 4.4 = 0.25
Therefore, the probability that an elevator arrives in less than 1.1 minutes is 0.250.
c. To calculate the probability that the wait for an elevator is more than 1.1 minutes, we can subtract the probability found in part (b) from 1.
Probability = 1 - Probability of less than 1.1 minutes
Probability = 1 - 0.250
Probability = 0.750
Therefore, the probability that the wait for an elevator is more than 1.1 minutes is 0.750.