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A man firing at a distant target has 20% chance of hitting the target in one shot. If P is the probability of hitting the target atleast once in 'nn' attempts and 20P2 − 13P + 2 ≤ 0, then the maximum value of 'n' is equal to
Correct answer is '2'. Can you explain this answer?
Most Upvoted Answer
A man firing at a distant target has 20% chance of hitting the target...
The probability of hitting the target in one shot = 1/20 = 1/5
The probability of not hitting the target in one shot 
∴ The probability of not hitting the target in all the n attempts = (4/5)n
⇒ The probability of hitting the target atleast once in n attempts, 
Also, given that 20P2 − 13P + 2 ≤ 0
⇒ 20P2 − 8P − 5P + 2 ≤ 0
⇒ 4P(5P − 2) − (5P − 2) ≤ 0
(4P − 1)(5P −2) ≤ 0
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A man firing at a distant target has 20% chance of hitting the target...
Problem Analysis:
- The man has a 20% chance of hitting the target in one shot, which means he has an 80% chance of missing the target in one shot.
- We need to find the maximum value of 'n' such that the probability of hitting the target at least once in 'n' attempts is greater than or equal to a certain value.
Solution:
To solve this problem, we need to use the concept of probability and combinatorics.
Step 1: Define the Probability of Hitting the Target At Least Once
- Let's assume the probability of hitting the target at least once in 'n' attempts is P.
- The probability of missing the target in one shot is 80%, so the probability of missing the target in 'n' attempts is (0.8)^n.
- Therefore, the probability of hitting the target at least once in 'n' attempts is 1 - (0.8)^n.
Step 2: Solve the Inequality
- Given the inequality 20P^2 - 13P^2 ≤ 0, we can simplify it as follows:
- 7P^2 ≤ 0
- Since P^2 is always non-negative, the inequality holds true when P = 0.
Step 3: Calculate the Maximum Value of 'n'
- We need to find the maximum value of 'n' such that the probability of hitting the target at least once (P) is greater than or equal to 0.
- From Step 1, we know that P = 1 - (0.8)^n.
- Setting P = 0, we have:
- 0 = 1 - (0.8)^n
- (0.8)^n = 1
- n*log(0.8) = log(1)
- n = log(1)/log(0.8)
- Using a calculator, we find that n ≈ 1.73697.
Step 4: Determine the Maximum Integer Value of 'n'
- Since 'n' represents the number of attempts, it must be a positive integer.
- The maximum integer value of 'n' that satisfies the inequality is 2, as it is the largest integer less than 1.73697.
Therefore, the maximum value of 'n' is equal to 2.
- The man has a 20% chance of hitting the target in one shot, which means he has an 80% chance of missing the target in one shot.
- We need to find the maximum value of 'n' such that the probability of hitting the target at least once in 'n' attempts is greater than or equal to a certain value.
Solution:
To solve this problem, we need to use the concept of probability and combinatorics.
Step 1: Define the Probability of Hitting the Target At Least Once
- Let's assume the probability of hitting the target at least once in 'n' attempts is P.
- The probability of missing the target in one shot is 80%, so the probability of missing the target in 'n' attempts is (0.8)^n.
- Therefore, the probability of hitting the target at least once in 'n' attempts is 1 - (0.8)^n.
Step 2: Solve the Inequality
- Given the inequality 20P^2 - 13P^2 ≤ 0, we can simplify it as follows:
- 7P^2 ≤ 0
- Since P^2 is always non-negative, the inequality holds true when P = 0.
Step 3: Calculate the Maximum Value of 'n'
- We need to find the maximum value of 'n' such that the probability of hitting the target at least once (P) is greater than or equal to 0.
- From Step 1, we know that P = 1 - (0.8)^n.
- Setting P = 0, we have:
- 0 = 1 - (0.8)^n
- (0.8)^n = 1
- n*log(0.8) = log(1)
- n = log(1)/log(0.8)
- Using a calculator, we find that n ≈ 1.73697.
Step 4: Determine the Maximum Integer Value of 'n'
- Since 'n' represents the number of attempts, it must be a positive integer.
- The maximum integer value of 'n' that satisfies the inequality is 2, as it is the largest integer less than 1.73697.
Therefore, the maximum value of 'n' is equal to 2.
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A man firing at a distant target has 20% chance of hitting the target in one shot. If P is the probability of hitting the target atleast once in 'nn' attempts and 20P2 − 13P + 2 ≤ 0, then the maximum value of 'n' is equal toCorrect answer is '2'. Can you explain this answer?
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A man firing at a distant target has 20% chance of hitting the target in one shot. If P is the probability of hitting the target atleast once in 'nn' attempts and 20P2 − 13P + 2 ≤ 0, then the maximum value of 'n' is equal toCorrect answer is '2'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A man firing at a distant target has 20% chance of hitting the target in one shot. If P is the probability of hitting the target atleast once in 'nn' attempts and 20P2 − 13P + 2 ≤ 0, then the maximum value of 'n' is equal toCorrect answer is '2'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A man firing at a distant target has 20% chance of hitting the target in one shot. If P is the probability of hitting the target atleast once in 'nn' attempts and 20P2 − 13P + 2 ≤ 0, then the maximum value of 'n' is equal toCorrect answer is '2'. Can you explain this answer?.
A man firing at a distant target has 20% chance of hitting the target in one shot. If P is the probability of hitting the target atleast once in 'nn' attempts and 20P2 − 13P + 2 ≤ 0, then the maximum value of 'n' is equal toCorrect answer is '2'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A man firing at a distant target has 20% chance of hitting the target in one shot. If P is the probability of hitting the target atleast once in 'nn' attempts and 20P2 − 13P + 2 ≤ 0, then the maximum value of 'n' is equal toCorrect answer is '2'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A man firing at a distant target has 20% chance of hitting the target in one shot. If P is the probability of hitting the target atleast once in 'nn' attempts and 20P2 − 13P + 2 ≤ 0, then the maximum value of 'n' is equal toCorrect answer is '2'. Can you explain this answer?.
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