In certain experiments, the error made in determining the density of a substance is a random variable having a uniform density with a=(-0.015) and b=0.015. Find the probability that such an error will be between (-0.002) and 0.003.

We are given that the error in determining the density of a substance is a random variable with a uniform distribution between \( a = -0.015 \) and \( b = 0.015 \). The probability density function (PDF) for a uniform distribution is given by:

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

In this case:

\[
f(x) = \frac{1}{0.015 - (-0.015)} = \frac{1}{0.03} = 33.33
\]

We are asked to find the probability that the error will be between \( -0.002 \) and \( 0.003 \). For a uniform distribution, the probability of the random variable falling between two values \( x_1 \) and \( x_2 \) is proportional to the length of the interval:

\[
P(x_1 \leq X \leq x_2) = \frac{x_2 - x_1}{b - a}
\]

Here, \( x_1 = -0.002 \) and \( x_2 = 0.003 \), so:

\[
P(-0.002 \leq X \leq 0.003) = \frac{0.003 - (-0.002)}{0.015 - (-0.015)} = \frac{0.003 + 0.002}{0.03} = \frac{0.005}{0.03} = 0.1667
\]

Thus, the probability that the error will be between \( -0.002 \) and \( 0.003 \) is **0.1667**, or 16.67%.

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. What is the probability that the error is between 0 and 0.01?
2. How would the result change if the interval for the uniform distribution were larger, say from -0.02 to 0.02?
3. What is the expected value of the error in this uniform distribution?
4. Calculate the variance of the error for the given uniform distribution.
5. How would the result change if the probability distribution was not uniform but normal?

**Tip**: In a uniform distribution, probabilities are proportional to the length of the interval you're considering!

In certain experiments, the error made in determining the density of a substance is a random variable having a uniform density with a = -0.015 and b = 0.015. Find the probability that such an error will be between -0.002 and 0.003.

Find the probability that the error in determining the density of a substance, modeled by a uniform distribution between -0.015 and 0.015, falls between -0.002 and 0.003. This problem covers key concepts in probability and uniform distributions, suitable for students in Grades 11-12.

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