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
B
7
C
9
D
11

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 hittingWe 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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