2019 Mock AMC 10B Problems/Problem 23
To solve this problem, we can consider each unit of to be separated into equal increments. (For example, if each unit of is separated into equal increments, there would be tiny fragments of per unit of .) Recall that , so we have to calculate (1) the number of ways to arrange “blocks” of in the whole “grid” of length given increments per unit (this is because the maximum least value for one of the sets is ) such that no two of them overlap and (2) the number of ways to arrange “blocks” of in the whole “grid” of length given increments per unit under no restrictions. We can easily see that the probability is
.
(The is included due to the four blocks being able to be arranged order.)
Since there are infinite real numbers in a unit of , there are infinite increments in a unit of , so we should take the limit as approaches infinity:
Dividing leading coefficients, we get the probability to be , which simplifies to .
Thus, our answer is .