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Question

A fireman is firing at a distant target and has only $$10\%$$ chance of hitting it. The number of rounds, he must fire in order to have $$50\%$$ chance of hitting it at least once is


A
5
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B
7
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C
9
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D
11
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Solution

The correct option is B $$7$$
For the first time the chance of hitting is $$\cfrac{1}{10}$$.
For the second time the chance of hitting is $$\cfrac{9}{10} \times \cfrac{1}{10}$$.
Let us assume that it takes $$x$$ rounds to get $$50$$% chance of hitting
We have $$\cfrac{1}{10}+\cfrac{9}{10} \times \cfrac{1}{10}+.........+(\cfrac{9}{10})^{(x-1)} \times \cfrac{1}{10} \ge \cfrac{1}{2}$$
$$\Rightarrow \cfrac{1}{10}(c\cfrac{9}{10}(\cfrac{1-(\cfrac{9}{10})^{(x-1)}}{1-\cfrac{9}{10}}))\ge\cfrac{1}{2}$$
$$\Rightarrow \cfrac{9}{10}-(\cfrac{9}{10})^{x} \ge \cfrac{1}{2}$$
Therefore for $$x=7$$ , there will be $$50$$% chance of hitting it atleast once.
Therefore the correct option is $$B$$.

Maths

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