New question. Suppose you sample one value from a uniform distribution with a = 0 and b = 50. What is the probability that the value will be between 30 and 46

To solve this problem, we'll calculate the probability that a randomly selected value from a uniform distribution between \(a = 0\) and \(b = 50\) falls between \(30\) and \(46\).

### Step 1: Understanding the Uniform Distribution
A uniform distribution between \(a = 0\) and \(b = 50\) means that every value within this interval is equally likely to be chosen. The probability density function (PDF) for a uniform distribution is given by:

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

Here, \(f(x)\) is constant across the interval \([0, 50]\).

### Step 2: Compute the PDF
Since \(a = 0\) and \(b = 50\), the PDF \(f(x)\) is:

\[
f(x) = \frac{1}{50-0} = \frac{1}{50}
\]

### Step 3: Calculate the Probability
The probability that a value \(X\) falls between 30 and 46 is the area under the PDF from \(x = 30\) to \(x = 46\). This can be found using the following integral:

\[
P(30 \leq X \leq 46) = \int_{30}^{46} f(x) \, dx
\]

Since \(f(x) = \frac{1}{50}\) is constant, the integral simplifies to:

\[
P(30 \leq X \leq 46) = \frac{1}{50} \times (46 - 30) = \frac{1}{50} \times 16 = \frac{16}{50} = 0.32
\]

### Final Answer
The probability that the value will be between 30 and 46 is \(0.32\) or 32%.

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Would you like more details or have any questions? Here are five related questions you might find interesting:

1. What is the probability of a value being less than 10 in the same distribution?
2. How would the probability change if the range was between 20 and 40 instead?
3. What is the mean and variance of this uniform distribution?
4. Can you calculate the probability that a value will be exactly 30?
5. How does the cumulative distribution function (CDF) relate to this problem?

**Tip:** In a uniform distribution, the probability of an interval is directly proportional to the length of that interval.

Learn how to calculate the probability that a value sampled from a uniform distribution with parameters a=0 and b=50 falls between 30 and 46. This problem explores the concept of uniform distribution probability density function and its application. Suitable for college-level students and learners interested in probability theory.

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