Question

Suppose that one variable, x, is chosen randomly and uniformly from [0, 3], and another variable, y, is also chosen randomly and uniformly from [0, 6]. What is the probability that xa) 1/4

b) 1/3
c) 1/2
d) 2/3

Answer 1

To find the probability, calculate the ratio of the area of the region where x satisfies the condition to the total area of the joint probability distribution. For x = 1/4, the probability is 1/12. For x = 1/3, the probability is 1/9. For x = 1/2, the probability is 1/6. For x = 2/3, the probability is 2/9.

Step-by-step explanation:

To find the probability, we need to determine the area of the region in the joint probability distribution where the values of x and y satisfy the given condition. In this case, we want to find the probability that x = 1/4, x = 1/3, x = 1/2, and x = 2/3. Since the variables x and y are chosen randomly and uniformly, the probability is equal to the ratio of the area of the region that satisfies the condition to the total area of the joint probability distribution.

Since x and y are both chosen randomly and uniformly, the joint probability distribution is a rectangle with sides [0, 3] and [0, 6]. The area of the rectangle is 3 * 6 = 18.

The area of the region where x = 1/4 is (1/4) * 6 = 1.5, so the probability is 1.5 / 18 = 1/12.

The area of the region where x = 1/3 is (1/3) * 6 = 2, so the probability is 2 / 18 = 1/9.

The area of the region where x = 1/2 is (1/2) * 6 = 3, so the probability is 3 / 18 = 1/6.

The area of the region where x = 2/3 is (2/3) * 6 = 4, so the probability is 4 / 18 = 2/9.