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. A number is selected at random from the first 1000 natural numbers. What is the probability that the number so selected would be a multiple of 7 or 11?
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. A number is selected at random from the first 1000 natural numbers. ...
Understanding the Problem
To find the probability of selecting a number from the first 1000 natural numbers that is a multiple of 7 or 11, we can use the principle of inclusion-exclusion.
Step 1: Identify Multiples of 7
- The multiples of 7 up to 1000 are given by:
- 7, 14, 21, ..., 994.
- To find how many multiples there are, we calculate:
- Last multiple = 7 * 142 = 994.
- Thus, there are 142 multiples of 7.
Step 2: Identify Multiples of 11
- The multiples of 11 up to 1000 are:
- 11, 22, 33, ..., 990.
- To find how many multiples there are:
- Last multiple = 11 * 90 = 990.
- Thus, there are 90 multiples of 11.
Step 3: Identify Common Multiples (Multiples of 77)
- The least common multiple of 7 and 11 is 77.
- The multiples of 77 up to 1000 are:
- 77, 154, 231, ..., 924.
- To find how many multiples there are:
- Last multiple = 77 * 12 = 924.
- Thus, there are 12 multiples of 77.
Step 4: Apply Inclusion-Exclusion Principle
- Total multiples of 7 or 11:
- Total = (Multiples of 7) + (Multiples of 11) - (Common multiples)
- Total = 142 + 90 - 12 = 220.
Step 5: Calculate Probability
- The total number of natural numbers from 1 to 1000 is 1000.
- Probability = (Total multiples of 7 or 11) / (Total numbers)
- Probability = 220 / 1000 = 0.22.
Conclusion
The probability that a randomly selected number from the first 1000 natural numbers is a multiple of 7 or 11 is 0.22 or 22%.
To find the probability of selecting a number from the first 1000 natural numbers that is a multiple of 7 or 11, we can use the principle of inclusion-exclusion.
Step 1: Identify Multiples of 7
- The multiples of 7 up to 1000 are given by:
- 7, 14, 21, ..., 994.
- To find how many multiples there are, we calculate:
- Last multiple = 7 * 142 = 994.
- Thus, there are 142 multiples of 7.
Step 2: Identify Multiples of 11
- The multiples of 11 up to 1000 are:
- 11, 22, 33, ..., 990.
- To find how many multiples there are:
- Last multiple = 11 * 90 = 990.
- Thus, there are 90 multiples of 11.
Step 3: Identify Common Multiples (Multiples of 77)
- The least common multiple of 7 and 11 is 77.
- The multiples of 77 up to 1000 are:
- 77, 154, 231, ..., 924.
- To find how many multiples there are:
- Last multiple = 77 * 12 = 924.
- Thus, there are 12 multiples of 77.
Step 4: Apply Inclusion-Exclusion Principle
- Total multiples of 7 or 11:
- Total = (Multiples of 7) + (Multiples of 11) - (Common multiples)
- Total = 142 + 90 - 12 = 220.
Step 5: Calculate Probability
- The total number of natural numbers from 1 to 1000 is 1000.
- Probability = (Total multiples of 7 or 11) / (Total numbers)
- Probability = 220 / 1000 = 0.22.
Conclusion
The probability that a randomly selected number from the first 1000 natural numbers is a multiple of 7 or 11 is 0.22 or 22%.
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. A number is selected at random from the first 1000 natural numbers. What is the probability that the number so selected would be a multiple of 7 or 11?
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. A number is selected at random from the first 1000 natural numbers. What is the probability that the number so selected would be a multiple of 7 or 11? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about . A number is selected at random from the first 1000 natural numbers. What is the probability that the number so selected would be a multiple of 7 or 11? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for . A number is selected at random from the first 1000 natural numbers. What is the probability that the number so selected would be a multiple of 7 or 11?.
. A number is selected at random from the first 1000 natural numbers. What is the probability that the number so selected would be a multiple of 7 or 11? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about . A number is selected at random from the first 1000 natural numbers. What is the probability that the number so selected would be a multiple of 7 or 11? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for . A number is selected at random from the first 1000 natural numbers. What is the probability that the number so selected would be a multiple of 7 or 11?.
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