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The probability that a number selected at random between 100 and 999 (both inclusive) will not contain the digit 3 is
- a)16/25
- b)9/10
- c)(9/10)2
- d)none of these
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The probability that a number selected at random between 100 and 999 (...
To find the probability that a randomly selected number between 100 and 999 does not contain the digit 3, follow these steps:
Count the total number of possible numbers:
Numbers between 100 and 999 inclusive are three-digit numbers. The total number of these numbers is:
999−100+1=900
Calculate the number of favorable outcomes (numbers without the digit 3):
Each digit of the number can be 0, 1, 2, 4, 5, 6, 7, 8, or 9, excluding 3. There are 9 choices for each digit.
For the hundreds place, the digit cannot be 0 (as it would not be a three-digit number), so it can be 1, 2, 4, 5, 6, 7, 8, or 9, giving us 8 choices.
For the tens and units places, the digit can be any of the 9 choices (0 through 9 excluding 3).
Therefore, the number of valid numbers is:
8×9×9=648
Calculate the probability:
The probability that a number does not contain the digit 3 is the ratio of the number of favorable outcomes to the total number of possible outcomes:
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The probability that a number selected at random between 100 and 999 (...
Understanding the Problem
To find the probability that a number selected at random between 100 and 999 does not contain the digit 3, we first need to understand the total possibilities and how many of those contain the digit 3.
Total Numbers in Range
- The range from 100 to 999 includes:
- Total numbers = 999 - 100 + 1 = 900
Counting Numbers Without the Digit 3
- **Hundreds Place**: The hundreds digit can be 1, 2, 4, 5, 6, 7, 8, or 9 (8 options).
- **Tens Place**: The tens digit can be 0, 1, 2, 4, 5, 6, 7, 8, or 9 (9 options).
- **Units Place**: The units digit can also be 0, 1, 2, 4, 5, 6, 7, 8, or 9 (9 options).
Calculating the Total Valid Combinations
- The total valid combinations without the digit 3 can be calculated as follows:
- Total combinations = (Choices for hundreds) × (Choices for tens) × (Choices for units)
- Total combinations = 8 × 9 × 9 = 648
Finding the Probability
- The probability that a number selected does not contain the digit 3 is given by:
- Probability = (Number of favorable outcomes) / (Total outcomes)
- Probability = 648 / 900
- Simplifying this gives:
- Probability = 72 / 100 = 4 / 25
Conclusion
- The answer choices provided do not match with 4/25, hence the correct answer is option 'D': none of these.
To find the probability that a number selected at random between 100 and 999 does not contain the digit 3, we first need to understand the total possibilities and how many of those contain the digit 3.
Total Numbers in Range
- The range from 100 to 999 includes:
- Total numbers = 999 - 100 + 1 = 900
Counting Numbers Without the Digit 3
- **Hundreds Place**: The hundreds digit can be 1, 2, 4, 5, 6, 7, 8, or 9 (8 options).
- **Tens Place**: The tens digit can be 0, 1, 2, 4, 5, 6, 7, 8, or 9 (9 options).
- **Units Place**: The units digit can also be 0, 1, 2, 4, 5, 6, 7, 8, or 9 (9 options).
Calculating the Total Valid Combinations
- The total valid combinations without the digit 3 can be calculated as follows:
- Total combinations = (Choices for hundreds) × (Choices for tens) × (Choices for units)
- Total combinations = 8 × 9 × 9 = 648
Finding the Probability
- The probability that a number selected does not contain the digit 3 is given by:
- Probability = (Number of favorable outcomes) / (Total outcomes)
- Probability = 648 / 900
- Simplifying this gives:
- Probability = 72 / 100 = 4 / 25
Conclusion
- The answer choices provided do not match with 4/25, hence the correct answer is option 'D': none of these.
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The probability that a number selected at random between 100 and 999 (both inclusive) will not contain the digit 3 isa)16/25b)9/10c)(9/10)2d)none of theseCorrect answer is option 'D'. Can you explain this answer?
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