Question

A can take a step forward with probability $0.4$ and backward with probability $0.6$. find the probability that at the end of eleven steps he is one step away from the starting point.

• A. probability$=0.57$
• B. probability$=0.63$
• C. probability$=0.43$
• D. probability$=0.37$

Answer 1

The correct option is D probability$=0.37$

Probability of going forward$=P\left(F\right)=0.4$
Probability of going backward$=P\left(B\right)=0.6$
After $11$ steps if he has to be one step ahead of starting point,then he must take in total $6$ steps ahead and $5$ backwards. This can be done in ${}^{11}{C}_6$ ways.
Hence,$P\left(Forward\right)={}^{11}{C}_6{P\left(F\right)}^6{P\left(B\right)}^5$
After $11$ steps if he has to be one step behind the starting point,then he must take in total $5$ steps ahead and $6$ backwards. This can be done in ${}^{11}{C}_5$ ways.
Hence,$P\left(Backward\right)={}^{11}{C}_5{P\left(F\right)}^5{P\left(B\right)}^6$

Total probability $=P\left(Forward\right)+P\left(Backward\right)=0.37$