Question

Two integers \(r\) and \(s\) are drawn one at a time without replacement from the set \(\{1, 2, \ldots, n\}\). Then \(P(r \leq k / s \leq k)\) is:

• A. \(\frac{k}{n}\)
• B. \(\frac{k}{n-1}\)
• C. \(\frac{k-1}{n}\)
• D. \(\frac{k-1}{n-1}\)

Hint:

When selecting from a set without replacement, consider the total number of ways to select and the number of favorable outcomes, then calculate the probability.

Answer 1

The Correct Option is D

Step 1: The question seeks the probability of \(r \leq s \leq k\), where \(r\) and \(s\) are drawn from \(\{1, 2, \dots, n\}\).

Step 2: Determine the total number of ways to select two integers from \(\{1, 2, \dots, n\}\): \(\binom{n}{2}\).

Step 3: Calculate the number of successful outcomes where \(r \leq s \leq k\). This equals \(k - 1\), as \(r\) and \(s\) are limited by \(k\) and their order.

Step 4: Compute the probability by dividing favorable outcomes by total outcomes, resulting in \(\frac{k-1}{n-1}\).