Answer: There is an infinite chessboard, and an ant A is in the middle of one of the squares.
The ant can move in any of the eight directions, from the center of one square to another. If it moves 1 square north, south, east or west; it requires 1 unit energy. If it moves to one of its diagonal neighbor (NE, NW, SE, SW); it requires 2–√ units of energy. It is equally likely to move in any of the eight directions. If it initially has 20 units of energy, find the probability that, after using the maximum possible energy, the ant will be 2 units away from its initial position.
I approached this problem, considering that the case that it finally ends up 2 units to the east (we can multiply by four to get all the cases).
If it ends up 2 units to the east, then Total steps to right =2+Total steps to left.
We will somehow balance these steps, considering that the ant has a total of 20 units of energy at the start.
I don't know how to effectively calculate the sample space either.
If the ant takes a total of n steps, such that while taking all n steps it is equally likely to move in any of the eight directions, then the sample space would be 8n.
But here we do not know n. Further, if the energy left after the second-last step is less than 2–√ and more than 1, then the ant will not be able to move diagonally.
I wasn't able to think of much after this. Help is appreciated.
Explanation: