The probability that X is greater than 0.8 is 0.36 or 36%.
The density curve of X forms a triangular shape with the base from 0 to 2, and its maximum height occurs at X = 1.
The probability that X is greater than 0.8 corresponds to the area of the triangle to the right of X = 0.8.
The area of a triangle is given by the formula:
![\[ \text{Area} = (1)/(2) * \text{base} * \text{height} \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/8kc38myn2hteh67zr2yf8pic2l519uvxia.png)
In this case, the base of the triangle is 2 and the height is 1 (as the maximum height is at X = 1). Therefore, the total area of the triangle is:
![\[ \text{Area} = (1)/(2) * 2 * 1 = 1 \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/vl024ou8k86xu8rcnme17nz98j96staxgv.png)
To find the probability that X is greater than 0.8, we need to find the ratio of the area to the right of X = 0.8 to the total area of the triangle.
The area to the right of X = 0.8 is a smaller triangle with a base of 1.2 (from 0.8 to 2) and a height of 0.6 (since it's a triangle within the larger triangle).
Therefore, the area to the right of X = 0.8 is:
![\[ \text{Area to the right of } X = 0.8 = (1)/(2) * 1.2 * 0.6 = 0.36 \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/pcjbnss3311ibgg8k16swgkrtp37j27pro.png)
Finally, the probability that X is greater than 0.8 is the ratio of this area to the total area of the triangle:
![\[ \text{Probability} = \frac{\text{Area to the right of } X = 0.8}{\text{Total Area}} = (0.36)/(1) = 0.36 \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/q07e8m80u0q3mguzsoorofzpg66e48logh.png)