A man takes a step forward with probability 0.4 and backward with… - Vidyadip

MCQ

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

✓
$^{11}{C_6}\,{(0.24)^5}$
B
$^{11}{C_6}{(0.4)^6}{(0.6)^5}$
C
$^{11}{C_6}{(0.6)^6}\,{(0.4)^5}$
D
None of these

✓

Answer

Correct option: A.

$^{11}{C_6}\,{(0.24)^5}$

a
(a) The man will be one step away from the starting point if $(i)$ either he is one step ahead or $(ii)$ one step behind the starting point.
$\therefore $The required probability $ = P(i) + P(ii)$
The man will be one step ahead at the end of eleven steps if he moves six steps forward and five steps backward. The probability of this event $ = \,{}^{11}{C_6}{(0.4)^6}{(0.6)^5}$.
The man will be one step behind at the end of eleven steps if he moves six steps backward and five steps forward.
The probability of this event $ = \,{}^{11}{C_6}{(0.6)^6}{(0.4)^5}$.
Hence the required probability
$ = \,{}^{11}{C_6}{(0.4)^6}{(0.6)^5} + {}^{11}{C_6}{(0.6)^6}{(0.4)^5}$
$ = \,{}^{11}{C_6}{(0.4)^5}{(0.6)^5}(0.4 + 0.6) = \,{}^{11}{C_6}{(0.24)^5}.$

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