ProbabilityHardBloom L3
Question
A man can take a step forward, backward, left or right with equal probability. Find the probability that after nine steps he will be just one step away from his initial position.
Options
A.$\frac{21347}{4^{9}}$
B.$\frac{3865}{4^{7}}$
C.$\frac{12345}{4^{9}}$
D.$\frac{3969}{4^{7}}$
Solution
Sol. [Key Concept: Random walk of 9 unit steps in 4 directions; count sequences ending one step from start.]
Case I $F = B$
| 0 | 0 | 4 | 5 |
|---|---|---|---|
| 1 | 1 | 3 | 4 |
| 2 | 2 | 2 | 3 |
| 3 | 3 | 1 | 2 |
| 4 | 4 | 0 | 1 |
Case II $L = R$
$$\begin{matrix} P & \ = \frac{4\left( \frac{9!}{4!5!} + \frac{9!}{3!4!1!1!} + \frac{9!}{2!2!2!3!} + \frac{9!}{3!3!1!2!} + \frac{9!}{4!4!1!} \right)}{4^{9}} \\ & \ = \frac{3969}{4} \end{matrix}$$
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