The lengths of a statistics lecture has a continuous uniform distribution between 50min and 52.0 min. If one such lecture of statistics is randomly selected, find the probability that the lecture length is more than 51 min.

Given that the length of a statistics lecture follows a continuous uniform distribution between 50 minutes and 52 minutes, we can model this using a uniform distribution \( U(a, b) \), where \( a = 50 \) minutes and \( b = 52 \) minutes.

### Step 1: Probability Density Function (PDF)
For a uniform distribution, the PDF is given by:

\[
f(x) = \frac{1}{b-a} \quad \text{for } a \leq x \leq b
\]

Substituting the values:

\[
f(x) = \frac{1}{52 - 50} = \frac{1}{2} = 0.5 \quad \text{for } 50 \leq x \leq 52
\]

### Step 2: Calculate the Probability
We need to find the probability that the lecture length is more than 51 minutes:

\[
P(X > 51) = \int_{51}^{52} f(x) \, dx
\]

Since \( f(x) = 0.5 \) for the entire interval \( 50 \leq x \leq 52 \), the probability is:

\[
P(X > 51) = \int_{51}^{52} 0.5 \, dx = 0.5 \times (52 - 51) = 0.5 \times 1 = 0.5
\]

### Final Answer
The probability that the lecture length is more than 51 minutes is \( \boxed{0.5} \) or 50%.

Would you like more details or have any questions?

Here are five related questions:
1. How do you calculate the mean of a uniform distribution?
2. What is the variance of a uniform distribution between two points \( a \) and \( b \)?
3. How would the probability change if the uniform distribution was between 50 minutes and 54 minutes?
4. What is the cumulative distribution function (CDF) for this uniform distribution?
5. How would you calculate the probability that the lecture length is between 50.5 and 51.5 minutes?

**Tip:** In a uniform distribution, the probability of an event occurring within a given interval is proportional to the length of that interval.

Calculate the probability that a statistics lecture selected from a uniform distribution between 50 minutes and 52 minutes exceeds 51 minutes. Understand the application of the probability density function (PDF) for uniform distributions with step-by-step examples.

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