Free Practice Question
PS · Problem Solving
sub-505
Probability
Consider a 3-digit integer chosen randomly from the range of 200 to 299, inclusive. What is the probability that the first digit and the last digit of this integer are each one greater than the middle digit?
Answer Choices
Correct answer marked belowA
2/225
B
1/111
C
1/110
1/100
Correct
E
1/50
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Explanation
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Official Explanation
Official
First, I need to understand the range and the condition:
Range: The numbers are from 200 to 299, inclusive. That means all 3-digit numbers where the first digit is 2.
Condition: For a number ABC (where A is the first digit, B is the middle digit, and C is the last digit), both A and C must be greater than B by exactly 1.
But since A is fixed as 2 (because the range is 200-299), the condition simplifies to:
A = B + 1 (but A is always 2, so B must be 1 because 2 = 1 + 1).
C = B + 1.
But C is the last digit, and B is the middle digit.
Step 1: Find Total Number of Possible Numbers
The range is from 200 to 299 inclusive. To find the total number of numbers:
Total numbers = 299 - 200 + 1 = 100.
Step 2: Understand the Condition in Detail
The number is in the form ABC, where:
A = 2 (since it's from 200 to 299).
B is the middle digit (0-9).
C is the last digit (0-9).
The condition says:
A > B by 1 → 2 = B + 1 → B = 1.
C > B by 1 → C = B + 1 → C = 1 + 1 = 2.
But, is it strictly greater, or greater by exactly one? The problem says "each one greater than the middle digit." It's a bit ambiguous.
Interpretation 1: A and C are both greater than B by exactly 1.
A = B + 1 → B = A - 1 = 2 - 1 = 1.
C = B + 1 = 1 + 1 = 2.
So, the number must be 212.
Interpretation 2: A and C are both greater than B, but not necessarily by exactly one.
A > B → 2 > B → B can be 0 or 1.
C > B → C > B.
But the problem says "one greater," which is more likely to mean "greater by one."
Given the wording, Interpretation 1 seems more accurate.
Verification
Looking at the exact wording: "the first digit and the last digit of this integer are each one greater than the middle digit."
"Each one greater" can be parsed as "each (first and last digit) is one greater than the middle digit."
Thus:
First digit (A) = B + 1 → B = A - 1 = 1.
Last digit (C) = B + 1 = 2.…
Range: The numbers are from 200 to 299, inclusive. That means all 3-digit numbers where the first digit is 2.
Condition: For a number ABC (where A is the first digit, B is the middle digit, and C is the last digit), both A and C must be greater than B by exactly 1.
But since A is fixed as 2 (because the range is 200-299), the condition simplifies to:
A = B + 1 (but A is always 2, so B must be 1 because 2 = 1 + 1).
C = B + 1.
But C is the last digit, and B is the middle digit.
Step 1: Find Total Number of Possible Numbers
The range is from 200 to 299 inclusive. To find the total number of numbers:
Total numbers = 299 - 200 + 1 = 100.
Step 2: Understand the Condition in Detail
The number is in the form ABC, where:
A = 2 (since it's from 200 to 299).
B is the middle digit (0-9).
C is the last digit (0-9).
The condition says:
A > B by 1 → 2 = B + 1 → B = 1.
C > B by 1 → C = B + 1 → C = 1 + 1 = 2.
But, is it strictly greater, or greater by exactly one? The problem says "each one greater than the middle digit." It's a bit ambiguous.
Interpretation 1: A and C are both greater than B by exactly 1.
A = B + 1 → B = A - 1 = 2 - 1 = 1.
C = B + 1 = 1 + 1 = 2.
So, the number must be 212.
Interpretation 2: A and C are both greater than B, but not necessarily by exactly one.
A > B → 2 > B → B can be 0 or 1.
C > B → C > B.
But the problem says "one greater," which is more likely to mean "greater by one."
Given the wording, Interpretation 1 seems more accurate.
Verification
Looking at the exact wording: "the first digit and the last digit of this integer are each one greater than the middle digit."
"Each one greater" can be parsed as "each (first and last digit) is one greater than the middle digit."
Thus:
First digit (A) = B + 1 → B = A - 1 = 1.
Last digit (C) = B + 1 = 2.…
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Comments
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P
PatrickJun 12, 2024
Great
I
IanJan 15, 2025
Thanks for sharingThank you for sharing!!thanks for sharing
What this question tests
What it tests
Your ability to compute probabilities of single and compound events, including 'at least one' questions.
Common trap
Forgetting to use the complement for 'at least one', or treating dependent events as independent.
Details
Difficulty
sub-505
Type
PS
Category