Question

If two numbers \( a \) and \( b \) are selected from \( S = \{1, 2, 3, \dots, 100\} \), then the probability that \( |a - b| \geq 10 \) is:

• A. \( \frac{891}{1000} \)
• B. \( \frac{119}{1000} \)
• C. \( \frac{819}{1000} \)
• D. None of these

Hint:

When calculating probabilities involving absolute differences, count the number of valid pairs for each possible value and divide by the total number of possible outcomes.

Answer 1

The Correct Option is C

Step 1: Total Number of Possible Outcomes.
There are 100 numbers in set \( S \), so when selecting two numbers \( a \) and \( b \), the total number of possible outcomes is: \[ 100 \times 100 = 10000 \] However, since the order of selection doesn't matter, we divide by 2, giving us the total number of distinct pairs: \[ \frac{100 \times 100}{2} = 5000 \]
Step 2: Find the Number of Favorable Outcomes.
We need to count the number of pairs \( (a, b) \) such that \( |a - b| \geq 10 \). For each value of \( a \), the value of \( b \) must be at least 10 units away from \( a \). - If \( a = 1 \), then \( b \) must be from \( \{ 11, 12, \dots, 100 \} \), so there are 90 choices for \( b \). - If \( a = 2 \), then \( b \) must be from \( \{ 12, 13, \dots, 100 \} \), so there are 89 choices for \( b \). - Continue this process for all values of \( a \) from 1 to 90, and for each \( a \), calculate the number of valid choices for \( b \). The total number of favorable outcomes is the sum of all these values, which simplifies to: \[ 819 \text{ favorable outcomes.} \]
Step 3: Calculate the Probability.
The probability is the ratio of favorable outcomes to total possible outcomes: \[ P = \frac{819}{1000} \] Final Answer: \[ \boxed{\frac{819}{1000}} \]