91Ó°ÊÓ

Define the sample space as all the possible positions where the trucks can break down on the road. Since the road has a length of L, we can consider it as a line segment of length L. Thus, the trucks can break down at any point on the line segment. So, we can model the positions of the trucks as an ordered triplet (x1, x2, x3), where xi represents the distance of truck i from the left end of the road, and 0≤xi≤L.

In order to determine the total number of cases, we have to calculate the volume of the region in which the truck positions can vary. Since the positions of the trucks are confined within a length L, the region is a 3-dimensional cube with side length L. The volume of this cube is given by V_total = L^3.

We now have to find the cases where no two trucks are within a distance d of each other. To do this, let's consider the volume of the region where at least two trucks are within a distance d of each other. For simplicity, let's analyze only the case where trucks 1 and 2 are close to each other (i.e., less than distance d apart). The remaining cases can be analyzed similarly. By the triangular inequality, x2 ≤ x1 + d, x1 ≤ x2 + d, and x3 ≥ max(x1, x2) - d. These inequalities represent a triangular prism with height L-d, base length d, and base width d. Thus, the volume of this region is V_unfavorable = 1/2 (L-d) * d^2.

From the total number of cases, we have to subtract all the unfavorable cases which include the case when trucks 1 and 2 are close to each other, trucks 2 and 3 are close, and trucks 1 and 3 are close. However, in doing so, we should also be careful not to subtract the cases when all three trucks are close to each other more than once (since they're included in each case). Thus, we get V_favorable = V_total - 3 * V_unfavorable + V_all_close, where V_all_close is the volume of the region where all three trucks are close to each other. In this case, V_all_close is a tetrahedron with side lengths d/2, d/2, and d, and its volume is given by (sqrt(2)/12) * d^3. Therefore, V_favorable = L^3 - 3 * [1/2 (L-d) * d^2] + (sqrt(2)/12) * d^3.

To find the desired probability, we have to divide the number of favorable cases by the total number of cases. So, the probability that no two trucks are within a distance d of each other is given by: P = V_favorable / V_total = [L^3 - 3 * 1/2 (L-d) * d^2 + (sqrt(2)/12) * d^3] / L^3.