Question 1

On Saturdays, cars arrive at Sam Schmitt's Scrub and Shine Car Wash at the rate of 6 cars per fifteen-minute interval.The probability that less than 10 minutes will elapse between car arrivals is ___.
A)0.8465
B)0.9817
C)0.0183
D)0.1535
E)0.2125

Accepted Answer

We need to find the probability that less than 10 minutes will elapse between car arrivals. 
First, we need to convert the car arrival rate to arrivals per minute. 
6 cars per 15 minutes = (6/15) cars per minute = 0.4 cars per minute.

Let X be the time between two car arrivals (in minutes). We want to find P(X < 0.1).

X follows an exponential distribution with parameter λ = 0.4 (since it has a Poisson arrival rate of 0.4 cars per minute).

The probability density function (pdf) of X is given by:
f(x) = λe^(-λx)

The cumulative distribution function (cdf) of X is given by:
F(x) = P(X < x) = 1 - e^(-λx)

Therefore, 
P(X < 0.1) = F(0.1) = 1 - e^(-0.4*0.1) ≈ 0.9817

So, the answer is B.

Question 2

Employees arrive at a cafeteria according to a Poisson process at an average rate of 30 employees per hour.The probability that after one employee arrives, the next one will arrive at least 3 minutes later is _________.
A)0.223
B)0.202
C)0.183
D)0.162
E)0.143

Accepted Answer

The Poisson process implies that the time between arrivals follows an exponential distribution with parameter λ, where λ is the average rate of arrivals per unit time. Given λ = 30 employees per hour, or 0.5 employees per minute (since 30 employees/hour * 1 hour/60 minutes = 0.5 employees/minute), the probability that the time until the next arrival is at least t minutes is given by P(T ≥ t) = e^(-λt). For t = 3 minutes, P(T ≥ 3) = e^(-0.5*3) = e^(-1.5) ≈ 0.223.

Question 3

On Saturdays, cars arrive at Sam Schmitt's Scrub and Shine Car Wash at the rate of 6 cars per fifteen-minute interval.The probability that at least 2 minutes will elapse between car arrivals is ___.
A)0.0000
B)0.4493
C)0.1353
D)1.0000
E)1.0225

Accepted Answer

The probability that at least 2 minutes will elapse between car arrivals is the complement of the probability that no more than 1 minute will elapse between car arrivals. The probability that no more than 1 minute will elapse between car arrivals is the probability that 0 or 1 car arrives in a 1-minute interval. The probability that 0 cars arrive in a 1-minute interval is e^(-6/15) = 0.8905. The probability that 1 car arrives in a 1-minute interval is (6/15) * e^(-6/15) = 0.1095. Therefore, the probability that no more than 1 car arrives in a 1-minute interval is 0.8905 + 0.1095 = 1.0000. The probability that at least 2 minutes will elapse between car arrivals is 1 - 1.0000 = 0.0000.

Question 4

Inquiries arrive at a record message device according to a Poisson process at a rate of 15 inquiries per minute.The probability that it takes more than 12 seconds for the first inquiry to arrive is approximately _________
A)0.05
B)0.75
C)0.25
D)0.27
E)0.73

Accepted Answer

The answer of Inquiries arrive at a record message device...

Question 5

In a busy gas station, an average of 2.4 cars arrive every minute.What is the average time between arrivals?
A)0.42 minutes
B)2.08 minutes
C)5.24 minutes
D)2.40 minutes
E)5.00 minutes

Accepted Answer

The average time between arrivals is the reciprocal of the arrival rate, so 1 / 2.4 = 0.4167 minutes, which rounds to approximately 0.42 minutes.

Question 6

Employees arrive at a cafeteria according to a Poisson process at an average rate of 30 employees per hour.The average waiting time between employees' arrivals is _________ minutes.
A)0.03
B)0.05
C)0.5
D)1
E)2

Accepted Answer

2The average waiting time between arrivals in a Poisson process is the inverse of the average rate of arrival. Since the rate is 30 employees per hour, the average waiting time is 1/30 hours. Converting hours to minutes (1 hour = 60 minutes), we get (1/30) * 60 = 2 minutes.

Question 7

In a busy gas station, an average of 2.4 cars arrive every 5 minutes.What is the average time between arrivals?
A)0.42 minutes
B)2.08 minutes
C)5.24 minutes
D)2.40 minutes
E)5.00 minutes

Accepted Answer

To find the average time between arrivals, we need to divide the time by the number of arrivals. Therefore, we can solve for:

(5 minutes) ÷ (2.4 cars) ≈ 2.08 minutes

Therefore, the average time between arrivals is approximately 2.08 minutes per car. So, the correct answer is option B. To find the average time between arrivals, we need to take the reciprocal of the average rate of arrivals. This is because rate is defined as the number of arrivals per unit of time. So, in this case, the average time between arrivals is 5/2.4 = 2.08 minutes. Therefore, the best choice is (B).

Quiz 6: Continuous Distributions

On Saturdays, cars arrive at Sam Schmitt's Scrub and Shine Car Wash at the rate of 6 cars per fifteen-minute interval.The probability that less than 10 minutes will elapse between car arrivals is ___.

Employees arrive at a cafeteria according to a Poisson process at an average rate of 30 employees per hour.The probability that after one employee arrives, the next one will arrive at least 3 minutes later is _________.

On Saturdays, cars arrive at Sam Schmitt's Scrub and Shine Car Wash at the rate of 6 cars per fifteen-minute interval.The probability that at least 2 minutes will elapse between car arrivals is ___.

Inquiries arrive at a record message device according to a Poisson process at a rate of 15 inquiries per minute.The probability that it takes more than 12 seconds for the first inquiry to arrive is approximately _________

In a busy gas station, an average of 2.4 cars arrive every minute.What is the average time between arrivals?

Employees arrive at a cafeteria according to a Poisson process at an average rate of 30 employees per hour.The average waiting time between employees' arrivals is _________ minutes.

In a busy gas station, an average of 2.4 cars arrive every 5 minutes.What is the average time between arrivals?