Answer :
To solve this problem, we need to find the probability that the duration of a professor's class is between 51.1 and 51.7 minutes, given that the class duration is uniformly distributed between 50.0 and 52.0 minutes.
The uniform distribution is a type of probability distribution where all outcomes are equally likely within a certain interval. For a continuous uniform distribution with a range from [tex]\( a \)[/tex] to [tex]\( b \)[/tex], the probability density function is constant and the total probability is spread evenly across the interval [tex]\([a, b]\)[/tex].
Steps to find the probability:
1. Identify the interval of interest: We are interested in the duration being between 51.1 and 51.7 minutes.
2. Determine the total range of the distribution: The class duration is uniformly distributed from 50.0 to 52.0 minutes. This means the total range of the distribution is:
[tex]\[
b - a = 52.0 - 50.0 = 2.0 \text{ minutes}
\][/tex]
3. Calculate the length of the interval of interest: The interval we want to find the probability for is from 51.1 to 51.7 minutes. The length of this interval is:
[tex]\[
\text{Interval length} = 51.7 - 51.1 = 0.6 \text{ minutes}
\][/tex]
4. Find the probability of the interval: Since the distribution is uniform, the probability that the class duration falls within a specific interval is the ratio of the length of the interval of interest to the total length of the distribution range. So, the probability is:
[tex]\[
P(51.1 < X < 51.7) = \frac{\text{Interval length}}{b - a} = \frac{0.6}{2.0} = 0.3
\][/tex]
Therefore, the probability that the professor's class duration is between 51.1 and 51.7 minutes is 0.3, or 30%.
The uniform distribution is a type of probability distribution where all outcomes are equally likely within a certain interval. For a continuous uniform distribution with a range from [tex]\( a \)[/tex] to [tex]\( b \)[/tex], the probability density function is constant and the total probability is spread evenly across the interval [tex]\([a, b]\)[/tex].
Steps to find the probability:
1. Identify the interval of interest: We are interested in the duration being between 51.1 and 51.7 minutes.
2. Determine the total range of the distribution: The class duration is uniformly distributed from 50.0 to 52.0 minutes. This means the total range of the distribution is:
[tex]\[
b - a = 52.0 - 50.0 = 2.0 \text{ minutes}
\][/tex]
3. Calculate the length of the interval of interest: The interval we want to find the probability for is from 51.1 to 51.7 minutes. The length of this interval is:
[tex]\[
\text{Interval length} = 51.7 - 51.1 = 0.6 \text{ minutes}
\][/tex]
4. Find the probability of the interval: Since the distribution is uniform, the probability that the class duration falls within a specific interval is the ratio of the length of the interval of interest to the total length of the distribution range. So, the probability is:
[tex]\[
P(51.1 < X < 51.7) = \frac{\text{Interval length}}{b - a} = \frac{0.6}{2.0} = 0.3
\][/tex]
Therefore, the probability that the professor's class duration is between 51.1 and 51.7 minutes is 0.3, or 30%.
