Vole Crossing

Discrete Mathematics Level 5

Vivian the Vole is leading her brother Vitalis across a \(\dfrac{9\sqrt{2}}{2}\) meters wide road that runs east-west.

  • Both voles run at 3 meters per second
  • Vitalis follows the exact path that Vivian takes
  • Vitalis follows 1 second behind Vivian

Starting at the edge of the road, Vivian will decide at random to run 1 meter northwest or 1 meter northeast every \(\dfrac{1}{3}\) second (she is equally likely to choose either direction).

Unfortunately, Vitalis is legally blind, so he can see no further than 250 centimeters. He also has a very poor sense of direction, so if he ever loses sight of Vivian, he cannot make it across the road.

Let \(\dfrac{a}{b}\) be the probability that both voles make it across the road, where \(a\) and \(b\) are coprime positive integers.

What is \(a+b\)?


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