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
$={}^{11}{C}_5{\left(0.4\right)}^5{\left(0.6\right)}^6+{}^{11}{C}_6{\left(0.4\right)}^6{\left(0.6\right)}^5$
$={}^{11}{C}_5{\left(0.4\right)}^5{\left(0.6\right)}^5\left(0.6+0.4\right)$
$={}^{11}{C}_5{\left(0.24\right)}^5$
$=462\times {\left(0.24\right)}^5$