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A Rifleman is Firing at a Distant Target and Has Only 10% Chance of Hitting It.The Least Number of Rounds He Must Fire in Order to Have More than 50% Chance of Hitting It at Least Once is (A) 11 (B) 9 - CBSE (Science) Class 12 - Mathematics

ConceptBernoulli Trials and Binomial Distribution

Question

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

• 11

• 9

• 7

• 5

Solution

7
Let p=chance of hitting a distant target

$\Rightarrow$ p =10% or p= 0.1

$\Rightarrow q = 1 - 0 . 1 = 0 . 9$
$\text{ Let n be the least number of rounds } .$
$P(\text{ hitting atleast once} ) = P(X \geq 1)$
$\Rightarrow 1 - P(X = 0) \geq 50 \%$
$\Rightarrow 1 - P(X = 0) \geq 0 . 5$
$P(X = 0) \leq 0 . 5$
$\Rightarrow (0 . 9 )^n \leq 0 . 5$
$\text{ Taking } \text{ log on both the sides, we get}$
$n \text{ log } 0 . 9 \leq \log 0 . 5$
$\Rightarrow n \leq \frac{\log 0 . 5}{\log 0 . 9}$
$\Rightarrow n \leq 7 . 2$
$\text{ Therefore, 7 is the least number of rounds that he must fire in order }$
$\text{ to have more than 50 % chance of hitting the target at least once } .$

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Solution A Rifleman is Firing at a Distant Target and Has Only 10% Chance of Hitting It.The Least Number of Rounds He Must Fire in Order to Have More than 50% Chance of Hitting It at Least Once is (A) 11 (B) 9 Concept: Bernoulli Trials and Binomial Distribution.
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