Probability

Grid Walking

Rectangular Grid Walk - Walls

         

An ant crawling along the coordinate plane wishes to travel from the origin to (3,2)(3, \, -2) while avoiding a wall between (1,1)(1, \, -1) and (2,1)(2, \, -1). If each move the ant makes must be 11 unit right or down, then how many possible paths could the ant take?

In a game of basketball, the underdog's advantage says that whenever one team has one more point than the other, the losing team will score a point. Given that teams score one point at a time in this basketball game until one team has 88 points, in how many different ways could the game unfold?

Pablo is at the point (4,3)(4, \, 3) in the coordinate plane and is able to walk 11 unit left or down each step he takes. There is a wall between (2,1)(2, \, 1) and (2,2)(2, \, 2) and between (1,0)(1, 0) and (1,1)(1, \, 1). How many possible paths could Pablo take from (4,3)(4, \, 3) to the origin?

In a game of soccer, the home team advantage says that whenever the home team is one point away from winning, the other team is unable to score. Given that teams score one point at a time in a soccer game until one team has 55 points, in how many different ways could the game unfold?

Carmen San Diego is hiding in the coordinate plane at (5,3)(5, \, 3). She has a trap set between (3,2)(3, \, 2) and (4,2)(4, \, 2) and between (4,1)(4, \, 1) and (4,2)(4, \, 2). If you can start at the origin and can only move 11 unit right or up each turn, in how many ways can you get to her position?

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