Math Problem Statement
Let k and n be positive integers with k ≤ n. We have an urn containing n balls labeled 1, 2, . . . , n, and draw k balls without replacement uniformly at random. Find the expected value of the label of the smallest ball.
Solution
We are asked to find the expected value of the label of the smallest ball drawn when balls are drawn without replacement from an urn containing balls labeled . Let's work through the solution step-by-step.
1. Problem Setup
- There are labeled balls: .
- We draw balls without replacement.
- We are interested in the expected value of the smallest label among the chosen balls.
2. Key Insight: Distribution of the Smallest Ball
Let denote the label of the smallest ball drawn. Since the balls are drawn uniformly at random, each subset of size is equally likely. We aim to find the expected value of , which can be computed as:
3. Computing the Probability
For the smallest ball to have label , all other balls chosen must have labels from the set . The number of ways to choose balls from this set is:
Since there are ways to choose any balls from the balls, the probability that the smallest ball has label is:
4. Setting Up the Expected Value Formula
We now substitute the expression for into the expected value formula:
We can factor out the constant :
5. Simplifying the Sum
This sum can be evaluated using known combinatorial identities, but the key idea is that the final result is known to simplify to:
6. Final Answer
The expected value of the label of the smallest ball is:
Summary
The intuition behind the result is that the labels are evenly spread among the balls, and when drawing balls, the expected value of the smallest label corresponds to this balanced formula.
Would you like further clarifications or more details on any step?
Here are some related questions to explore:
- How would the result change if the draws were made with replacement?
- What is the variance of the smallest ball's label?
- How would the expected value change if we were interested in the largest label drawn?
- Can we generalize this result to continuous distributions (e.g., picking random real numbers)?
- How does the distribution of the smallest ball's label behave for large and small ?
Tip: Recognizing patterns in combinatorial sums is often key to finding elegant solutions to complex probability problems.
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Combinatorics
Expected Value
Formulas
E[X] = (n + 1) / (k + 1)
Pr(X = x) = binom(n - x, k - 1) / binom(n, k)
Theorems
Combinatorial Probability
Linear Expectation
Suitable Grade Level
College Level (Undergraduate Probability/Combinatorics)
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