To solve this problem, we need to find the probability of the shaded region under the uniform distribution.

Key information:

• The height of the distribution is $\frac{1}{9}$.
• The distribution is uniform, so the probability is proportional to the area of the shaded region.

Step-by-step solution:

1. Identify the length of the shaded region:

   • From the graph, the shaded region lies between $x = 6$ and $x = 8$.
   • The length of the shaded region is $8 - 6 = 2$.

2. Find the area of the shaded region:

   • The area of a region under the curve in a uniform distribution is given by: $\text{Area} = \text{length of region} \times \text{height of distribution}$
   • Substituting the values: $\text{Area} = 2 \times \frac{1}{9} = \frac{2}{9}$

3. Conclusion:

   • The probability of the shaded region is $\frac{2}{9}$.

This is the final answer: $\frac{2}{9}$.

Would you like further details, or have any questions? Here are five related questions that might interest you:

1. What is a uniform distribution and how is it defined mathematically?
2. How would the probability change if the shaded region extended from $x = 4$ to $x = 9$?
3. Can the probability ever exceed 1 in a probability distribution?
4. How is the total area under a probability density function related to probability?
5. What are the properties of continuous versus discrete uniform distributions?

Tip: The area under a probability density function (PDF) over a given interval represents the probability for that interval, and the total area under the PDF for the entire distribution equals 1.