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A man takes a step forward with probability $0.4$ and backward with probability $0.6$ . If the probability that at the end of eleven steps he is one step away from the starting point is $\lambda$ then the value of $5000 \lambda$ must be
Option: 1
$1580$

Option: 2
$1850$

Option: 3
$1058$

Option: 4
$5000$

Answers (1)

best_answer

Since, the man is one step away from starting point mean that either
(i) man has taken 6 steps forward and 5 steps backward.
(ii) man has taken 5 steps forward and 6 steps backward.
Taking, movement 1 step forward as success and 1 step backward as failure.

$\mathrm{\therefore \quad p=}$ probability of success $\mathrm{=0.4}$

and $\mathrm{q=}$ probability of failure $\mathrm{=0.6}$

$\mathrm{\therefore}$ Required probability $\mathrm{=P\{X=6 \: or \: X=5\}}$

$\mathrm{ =P(X=6)+P(X=5) }$

$\mathrm{ ={ }^{11} C_6 p^6 q^5+{ }^{11} C_5 p^5 q^6 }$

$\mathrm{ ={ }^{11} C_5\left(p^6 q^5+p^5 q^6\right) }$

$\mathrm{ =\frac{11 \cdot 10 \cdot 9 \cdot 8 \cdot 7}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} \left\{(0.4)^6(0.6)^5+(0.4)^5(0.6)^6\right\} }$

$\mathrm{ =\frac{11 \cdot 10 \cdot 9 \cdot 8 \cdot 7}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5}(0.24)^5 }$

$\mathrm{ =0.37}$

Hence, the required probability $\mathrm{=0.37=\lambda}$ (given)

$\mathrm{\therefore 5000 \lambda =5000 \times 0.37 }$

$=1850$

Hence option 2 is correct.

Posted by

Ritika Jonwal

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