Without An Appointment, The Average Waiting Time In Minutes At The Doctor's Office Has The Probability Density Function F ( T ) = 1 42 F(t)=\frac{1}{42} F ( T ) = 42 1 ​ , Where 0 ≤ T ≤ 42 0 \leq T \leq 42 0 ≤ T ≤ 42 .Step 1 Of 2: What Is The Probability That You Will Wait At Least 9

Introduction

Waiting in line at the doctor's office can be a frustrating experience, especially when you don't have an appointment. In this article, we will explore the probability density function (PDF) of waiting times at the doctor's office, which is given by $f(t)=\frac{1}{42}$, where $0 \leq t \leq 42$. Our goal is to determine the probability that you will wait at least 9 minutes.

Understanding the Probability Density Function

The probability density function (PDF) is a mathematical function that describes the probability distribution of a continuous random variable. In this case, the PDF $f(t)=\frac{1}{42}$ represents the probability density of waiting times at the doctor's office. The PDF is defined for the interval $0 \leq t \leq 42$, which means that the waiting time can range from 0 to 42 minutes.

Calculating the Probability

To calculate the probability that you will wait at least 9 minutes, we need to integrate the PDF over the interval $[9, 42]$. This can be done using the following formula:

$$P(t \geq 9) = \int_{9}^{42} f(t) dt$$

where $P(t \geq 9)$ is the probability that you will wait at least 9 minutes.

Solving the Integral

To solve the integral, we can use the following steps:

1. Evaluate the integral:

$$\int_{9}^{42} \frac{1}{42} dt = \frac{1}{42} \int_{9}^{42} dt$$

2. Simplify the integral:

$$\frac{1}{42} \int_{9}^{42} dt = \frac{1}{42} [t]_{9}^{42}$$

3. Evaluate the limits:

$$\frac{1}{42} [t]_{9}^{42} = \frac{1}{42} (42 - 9)$$

4. Simplify the expression:

$$\frac{1}{42} (42 - 9) = \frac{1}{42} \cdot 33$$

5. Simplify the fraction:

$$\frac{1}{42} \cdot 33 = \frac{33}{42}$$

6. Simplify the fraction further:

$$\frac{33}{42} = \frac{11}{14}$$

Conclusion

In conclusion, the probability that you will wait at least 9 minutes at the doctor's office is $\frac{11}{14}$. This means that there is a $\frac{11}{14}$ chance that you will wait at least 9 minutes, and a $\frac{3}{14}$ chance that you will wait less than 9 minutes.

Step 2: What is the probability that you will wait more than 21 minutes?

To calculate the probability that you will wait more than 21 minutes, we need to integrate the PDF over the interval $[21, 42]$. This can be done using the following formula:

$$P(t > 21) = \int_{21}^{42} f(t) dt$$

where $P(t > 21)$ is the probability that you will wait more than 21 minutes.

Solving the Integral

To solve the integral, we can use the following steps:

1. Evaluate the integral:

$$\int_{21}^{42} \frac{1}{42} dt = \frac{1}{42} \int_{21}^{42} dt$$

2. Simplify the integral:

$$\frac{1}{42} \int_{21}^{42} dt = \frac{1}{42} [t]_{21}^{42}$$

3. Evaluate the limits:

$$\frac{1}{42} [t]_{21}^{42} = \frac{1}{42} (42 - 21)$$

4. Simplify the expression:

$$\frac{1}{42} (42 - 21) = \frac{1}{42} \cdot 21$$

5. Simplify the fraction:

$$\frac{1}{42} \cdot 21 = \frac{21}{42}$$

6. Simplify the fraction further:

$$\frac{21}{42} = \frac{1}{2}$$

Conclusion

In conclusion, the probability that you will wait more than 21 minutes at the doctor's office is $\frac{1}{2}$. This means that there is a $\frac{1}{2}$ chance that you will wait more than 21 minutes, and a $\frac{1}{2}$ chance that you will wait 21 minutes or less.

Final Thoughts

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Frequently Asked Questions

Q: What is the probability density function of waiting times at the doctor's office? A: The probability density function of waiting times at the doctor's office is given by $f(t)=\frac{1}{42}$, where $0 \leq t \leq 42$.

Q: What is the probability that I will wait at least 9 minutes? A: The probability that you will wait at least 9 minutes is $\frac{11}{14}$.

Q: What is the probability that I will wait more than 21 minutes? A: The probability that you will wait more than 21 minutes is $\frac{1}{2}$.

Q: How long is the average waiting time at the doctor's office? A: The average waiting time at the doctor's office is 21 minutes.

Q: What is the probability that I will wait exactly 21 minutes? A: The probability that you will wait exactly 21 minutes is 0, since the waiting time is a continuous random variable.

Q: Can I use the probability density function to determine the waiting time for a specific appointment? A: No, the probability density function only provides information about the probability distribution of waiting times, not the actual waiting time for a specific appointment.

Q: How can I use the probability density function to make informed decisions about scheduling appointments? A: You can use the probability density function to determine the expected waiting time and the probability of waiting a certain amount of time. This can help you make informed decisions about scheduling appointments, such as choosing a time slot that is less likely to have a long wait.

Q: Is the probability density function the same for all doctor's offices? A: No, the probability density function is specific to the doctor's office and may vary depending on factors such as the number of patients, the number of doctors, and the efficiency of the office.

Q: Can I use the probability density function to compare the waiting times at different doctor's offices? A: Yes, you can use the probability density function to compare the waiting times at different doctor's offices. By comparing the probability density functions, you can determine which office has a shorter or longer average waiting time.

Q: How can I obtain the probability density function for a specific doctor's office? A: You can obtain the probability density function for a specific doctor's office by collecting data on waiting times and using statistical methods to estimate the probability density function.

Conclusion

In this article, we have answered some frequently asked questions about the probability density function of waiting times at the doctor's office. We have provided information about the probability density function, including its definition, the probability that you will wait at least 9 minutes, and the probability that you will wait more than 21 minutes. We have also discussed how to use the probability density function to make informed decisions about scheduling appointments and how to compare the waiting times at different doctor's offices.