Starting at $(0,0)$ an object moves in coordinate plane by a sequence of steps, each of length one.

Question

Starting at $(0,0)$ an object moves in coordinate plane by a sequence of steps, each of length one. Each step is left, right up or down, all four equally likely. Find the probability that the object reaches $(2,2)$ in six or fewer steps.

Accepted Answer

Sol. [Key Concept: Object moving by unit steps: count favorable lattice routes.]4 steps or 6 steps$$P = \frac{\frac{4!}{2!2!}}{4^{4}} + \frac{+ 2\left( \frac{6!}{312!1!} - \frac{4!}{2!2!} 	imes 2! ight)}{4^{6}} = \frac{3}{64}$$

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