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The probability that a k-digit number does NOT contain the digits 0.5, or 9 is
- a)0.3k
- b)0.6k
- c)0.7k
- d)0.9k
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
The probability that a k-digit number does NOT contain the digits 0.5,...
Each digit can be filled in 7 ways as 0, 5 and 9 is not allowed, so each of these places can be filled by 1, 2, 3, 4, 6, 7, 8
So required probability is 
Most Upvoted Answer
The probability that a k-digit number does NOT contain the digits 0.5,...
To find the probability that a k-digit number does not contain the digits 0, 5, or 9, we need to consider the total number of possible k-digit numbers and subtract the number of k-digit numbers that contain 0, 5, or 9.
Total Number of Possible k-Digit Numbers:
- Each digit in a k-digit number can take values from 0 to 9 (excluding 0, 5, and 9).
- Therefore, each digit has 10 - 3 = 7 possible choices.
- Since each digit is independent of the others, the total number of possible k-digit numbers is (7)^k.
Number of k-Digit Numbers that Contain 0, 5, or 9:
- For a k-digit number to contain 0, 5, or 9, each digit can take values from 0 to 9 (including 0, 5, and 9).
- Therefore, each digit has 10 possible choices.
- Since each digit is independent of the others, the total number of k-digit numbers that contain 0, 5, or 9 is (10)^k.
Subtracting the number of k-digit numbers that contain 0, 5, or 9 from the total number of possible k-digit numbers gives us the number of k-digit numbers that do not contain 0, 5, or 9.
Probability that a k-Digit Number Does Not Contain 0, 5, or 9:
- The probability is given by the ratio of the number of k-digit numbers that do not contain 0, 5, or 9 to the total number of possible k-digit numbers.
- So, the probability is ((7)^k - (10)^k) / (7)^k.
Simplifying the expression, we get (1 - (10/7)^k).
As k approaches infinity, (10/7)^k approaches 0 because the value is less than 1.
Therefore, the probability approaches 1 - 0 = 1.
Hence, the correct answer is option 'C' (0.7k), indicating that the probability that a k-digit number does not contain the digits 0, 5, or 9 is approximately 0.7k.
Total Number of Possible k-Digit Numbers:
- Each digit in a k-digit number can take values from 0 to 9 (excluding 0, 5, and 9).
- Therefore, each digit has 10 - 3 = 7 possible choices.
- Since each digit is independent of the others, the total number of possible k-digit numbers is (7)^k.
Number of k-Digit Numbers that Contain 0, 5, or 9:
- For a k-digit number to contain 0, 5, or 9, each digit can take values from 0 to 9 (including 0, 5, and 9).
- Therefore, each digit has 10 possible choices.
- Since each digit is independent of the others, the total number of k-digit numbers that contain 0, 5, or 9 is (10)^k.
Subtracting the number of k-digit numbers that contain 0, 5, or 9 from the total number of possible k-digit numbers gives us the number of k-digit numbers that do not contain 0, 5, or 9.
Probability that a k-Digit Number Does Not Contain 0, 5, or 9:
- The probability is given by the ratio of the number of k-digit numbers that do not contain 0, 5, or 9 to the total number of possible k-digit numbers.
- So, the probability is ((7)^k - (10)^k) / (7)^k.
Simplifying the expression, we get (1 - (10/7)^k).
As k approaches infinity, (10/7)^k approaches 0 because the value is less than 1.
Therefore, the probability approaches 1 - 0 = 1.
Hence, the correct answer is option 'C' (0.7k), indicating that the probability that a k-digit number does not contain the digits 0, 5, or 9 is approximately 0.7k.
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The probability that a k-digit number does NOT contain the digits 0.5, or 9 isa)0.3kb)0.6kc)0.7kd)0.9kCorrect answer is option 'C'. Can you explain this answer?
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