2019 AIME I Problems/Problem 5
A moving particle starts at the point and moves until it hits one of the coordinate axes for the first time. When the particle is at the point , it moves at random to one of the points , , or , each with probability , independently of its previous moves. The probability that it will hit the coordinate axes at is , where and are positive integers such that is not divisible by . Find .
One could recursively compute the probabilities of reaching as the first axes point from any point as for and the base cases are for any not equal to zero. We then recursively find so the answer is .
If this algebra seems intimidating, you can watch a nice pictorial explanation of this by On The Spot Stem. https://www.youtube.com/watch?v=XBRuy3_TM9w
Obviously, the only way to reach (0,0) is to get to (1,1) and then have a chance to get to (0,0). Let x denote a move left 1 unit, y denote a move down 1 unit, and z denote a move left and down one unit each. The possible cases for these moves are and . This gives a probability of to get to . The probability of reaching is . This gives .
Unique solution: https://youtu.be/I-8xZGhoDUY
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