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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

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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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