Question

Generate two random numbers between 0 and 1(all numbers equally likely to be chosen) and take X to be their sum. The sum X can take any value between 0 and 2. The density curve of X gives the triangle shown below. What is the probability that X is greater than 0.8? (Give an exact answer with no rounding.)

Answer 1

To calculate the probability that X is greater than 0.8, we need to find the area under the density curve of X to the right of 0.8. The density curve of X is a triangle with a base of 2 and a height of 1, centered at x = 1. Therefore, the probability that X is greater than 0.8 is 0.68.

Step-by-step explanation:

To calculate the probability that X is greater than 0.8, we need to find the area under the density curve of X to the right of 0.8. The density curve of X is a triangle with a base of 2 and a height of 1, centered at x = 1. So, the total area under the curve is (1/2)*(base)*(height) = (1/2)*(2)*(1) = 1.

The area to the left of 0.8 is (1/2)*(0.8)*(0.8) = 0.32.

Therefore, the probability that X is greater than 0.8 is 1 - P(X < 0.8) = 1 - 0.32 = 0.68.

Answer 2

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:

`\[ \text{Area} = (1)/(2) * \text{base} * \text{height} \]`

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:

`\[ \text{Area} = (1)/(2) * 2 * 1 = 1 \]`

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:

`\[ \text{Area to the right of } X = 0.8 = (1)/(2) * 1.2 * 0.6 = 0.36 \]`

Finally, the probability that X is greater than 0.8 is the ratio of this area to the total area of the triangle:

`\[ \text{Probability} = \frac{\text{Area to the right of } X = 0.8}{\text{Total Area}} = (0.36)/(1) = 0.36 \]`