Question

2. Uniform Distribution Two line segments, each of length two units, are placed along the x-axis. The midpoint of the first is between x=0 and x=11 and that of the second is between x=6 and x=22. Assuming independence and uniform distributions for these midpoints, find the probability that the line segments overlap. (10 points) SHOW REASONING

Answer 1

To find the probability that the line segments overlap, we need to consider the possible positions of the midpoints of the two line segments. The probability can be calculated by finding the overlap of the two uniform distributions. In this case, the probability is 2.75.

Step-by-step explanation:

To find the probability that the line segments overlap, we need to consider the possible positions of the midpoints of the two line segments. The midpoint of the first line segment can be anywhere between x=0 and x=11, and the midpoint of the second line segment can be anywhere between x=6 and x=22.

Since the distributions of the midpoints are uniform and independent, we can calculate the probability of the line segments overlapping by finding the overlap of the two uniform distributions. The length of the overlap will be the maximum of the minimum values of the two distributions minus the minimum of the maximum values of the two distributions.

In this case, the minimum value of the first distribution is 5 (midpoint of x=0 and x=11), and the maximum value of the second distribution is 16 (midpoint of x=6 and x=22). Therefore, the length of the overlap is 16 - 5 = 11 units.

The total possible length of the two line segments is 2 + 2 = 4 units. Therefore, the probability that the line segments overlap is 11/4 = 2.75.

Answer 2

The probability of overlap is 0.4545 or 45.45%.

Step-by-step explanation:

To find the probability that the line segments overlap, we need to determine the range of possible positions for the midpoints of the segments and calculate the probability of overlap within that range.

The first midpoint can be between the values of 0 and 11, and the second midpoint can be between 6 and 22.

To calculate the probability of overlap, we need to find the intersection between these ranges. The range of possible positions for the overlapping segment is the intersection between the two ranges, which is from 6 to 11.

The length of this range is 11 - 6 = 5 units.

The total range of possible positions for the midpoints of the segments is 11 - 0 = 11 units for the first segment and 22 - 6 = 16 units for the second segment.

Since both segments have the same length of 2 units, the probability of overlap is the ratio of the length of the overlapping range to the total range of possible positions.

Therefore, the probability of overlap is 5 / 11 = 0.4545 or 45.45%.