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The given problem states that \( X \) is the weight, in grams, of a gear used in the rotary mechanism of a microwave oven. The probability density function (pdf) is defined as:

\[
f(x) = \frac{1}{3}, \quad \text{for } 3.5 \text{ grams} < x < 6.5 \text{ grams}.
\]

You are asked to find the probability that the weight of one of these parts is less than 5.661 grams, i.e., \( P(X < 5.661) \), and round the answer to four decimal places.

Since this is a uniform distribution, the probability is calculated using:

\[
P(a < X < b) = \frac{b - a}{\text{length of the interval}}.
\]

The length of the interval is:

\[
6.5 - 3.5 = 3 \text{ grams}.
\]

We now need to find the probability that \( X \) is less than 5.661:

\[
P(X < 5.661) = \frac{5.661 - 3.5}{6.5 - 3.5} = \frac{5.661 - 3.5}{3}.
\]

Let me compute that.The probability that the weight of one of these parts is less than 5.661 grams is approximately **0.7203**.

Would you like more details or have any questions?

Here are some related questions to further explore the topic:
1. What is the probability that the weight is between 4.0 and 6.0 grams?
2. How would the calculation change if the distribution was not uniform but normal?
3. What is the expected value of the weight in this uniform distribution?
4. How can we compute the cumulative distribution function (CDF) for this problem?
5. What if the range of weights is extended to 3.0 grams to 7.0 grams?

**Tip:** In a uniform distribution, the probability is proportional to the interval length, making the calculations relatively straightforward.

X is the weight, in grams, of a gear used in the rotary mechanism of a microwave oven. The pdf is given as f(x) = 1/3 for 3.5 grams < x < 6.5 grams. The probability that the weight of one of these parts is less than 5.661 grams is _____. Answer to four decimal places.

This problem asks to calculate the probability that the weight of a microwave gear is less than 5.661 grams, given a uniform distribution with a probability density function f(x) = 1/3. Suitable for university students learning probability concepts.

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