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The question is about a uniform distribution of a random variable \( X \), and you are asked to find the probability \( P(X > 1.6) \) or \( P(X < 0.12) \) from the given uniform distribution over the interval [0, 2].

For a uniform distribution over an interval \([a, b]\), the probability density function (PDF) is given by:

\[
f(x) = \frac{1}{b - a}
\]

Here, \(a = 0\) and \(b = 2\), so the PDF is:

\[
f(x) = \frac{1}{2 - 0} = \frac{1}{2}
\]

The probability of \(X\) being in any subinterval is simply the length of that subinterval multiplied by the PDF. Thus, to compute the probabilities:

1. **\( P(X > 1.6) \):**

The length of the interval \( [1.6, 2] \) is:

\[
2 - 1.6 = 0.4
\]

So, the probability is:

\[
P(X > 1.6) = 0.4 \times \frac{1}{2} = 0.2
\]

2. **\( P(X < 0.12) \):**

The length of the interval \( [0, 0.12] \) is:

\[
0.12 - 0 = 0.12
\]

So, the probability is:

\[
P(X < 0.12) = 0.12 \times \frac{1}{2} = 0.06
\]

3. **For \( P(X > 1.6) \) or \( P(X < 0.12) \):**

Since these two events are mutually exclusive (they can't happen at the same time), you can add the probabilities:

\[
P(X > 1.6 \text{ or } X < 0.12) = P(X > 1.6) + P(X < 0.12) = 0.2 + 0.06 = 0.26
\]

Thus, the final answer is **0.26**.

Would you like more details or have any questions? Here are some related questions to consider:

1. How do we calculate the probability for a uniform distribution over different intervals?
2. What are the properties of a uniform distribution?
3. Can uniform distributions be discrete or only continuous?
4. How would the probability change if the interval for \(X\) were different, like \([0, 3]\)?
5. What are some real-world applications of uniform distributions?

**Tip:** Always visualize the interval on the number line for uniform distributions to better understand the probabilities associated with different regions.

From the figure, what is P(X > 1.6) or P(X < 0.12)?

Learn how to calculate probabilities for a uniform distribution on the interval [0, 2], focusing on finding P(X > 1.6) and P(X < 0.12). This guide covers the use of probability density functions and simple multiplication to derive probabilities in a continuous uniform distribution.

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