On a rod of length $6$ units, lengths $1, 2$ units are measured at random, the probability that no points of the measured lines will coincide is ______.
Step-by-Step Solution
Key Concept: When measuring lengths a and b from endpoints of a rod of length L, the measured points coincide when both measurements reach the same location. The favorable region (no coincidence) forms a rectangle with area c² where c = L - a - b, within total sample space (a+c)(b+c), yielding probability c²/[(a+c)(b+c)].
On a rod of length $a+b+c$, measure lengths $a, b$ from endpoints. The favorable region where no measured point coincides is a rectangle with area $c^2$ within a rectangle of total area $(a+c)(b+c)$. Therefore, the probability is $\frac{c^2}{(a+c)(b+c)}$.
Correct Answer: 0.45