Question

A drunken man takes a step forward with probability $0.4$ and backwards with probability $0.6$. Find the probability that at the end of eleven steps, he is one step away from the starting point.

• A. $24 \times (0.36)^6$
• B. $462 \times (0.24)^6$
• C. $24 \times (0.36)^5$
• D. $462 \times (0.24)^5$

Answer 1

Answer: (D)

The man is one step away from starting point after $11$ steps. This can happen in following two ways :
(i) he takes $5$ steps forward and $6$ steps backward
(ii) he takes $6$ steps forwards and $5$ steps backward
Probability in first case $= \,{}^{11}C_{5}\left(0.4\right)^{5}\,\left(0.6\right)^{6}$,
Probability in second case $= \,{}^{11}C_{6}\left(0.4\right)^{6}\,\left(0.6\right)^{5}$
Hence, required probability
$=\,{}^{11}C_{5}\left(0.4\right)^{5}\,\left(0.6\right)^{6}+\,{}^{11}C_{6}\left(0.4\right)^{6}\,\left(0.6\right)^{5}$
$= \,{}^{11}C_{5}\left(0.4\right)^{5}\,\left(0.6\right)^{5}\left(0.6+0.4\right)$
$= \,{}^{11}C_{5}\left(0.24\right)^{5}$
$= 462 \times \left(0.24\right)^{5}$