Question

A man takes a step forward with probability $0.4$ and backward with probability $0.6.$ Find the probability that at the end of $11$ steps he is $1$ step away from starting point

Answer 1

One step away means a step forward or a step backward. Either he took 6 forward steps and 5 backward steps or 6 backward steps and 5 forward steps.

Hence

Case 1

6 forward steps and 5 backward steps

${}^{11}{C}_6\times (0.4{)}^6\times (0.5{)}^5$ (1)

Case 2

5 forward steps and 6 backward steps

${}^{11}{C}_5\times (0.4{)}^5\times (0.5{)}^6$ (2)

Adding (1) and (2) gives requires probability$=0.368$