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Chapter 7: Problem 47

When \(m\) balls are distributed into \(n\) bins uniformly at random, what is the probability that the first bin remains empty?

Short Answer

Expert verified
The probability is \(\left(\frac{n-1}{n}\right)^m\).

Step by step solution

01

Understanding the Problem

We need to find the probability that the first bin remains empty when distributing balls into bins uniformly at random.
02

Identify Total Outcomes

Each ball can go into any of the bins. Therefore, the total number of ways to distribute the balls is \[ n^m \].
03

Calculate Favorable Outcomes

To keep the first bin empty, all balls must go into any of the remaining \(n-1\) bins. The number of ways this can happen is \[ (n-1)^m \].
04

Compute the Probability

The probability that the first bin remains empty is the ratio of the number of favorable outcomes to the total number of outcomes. Thus, \[ P(\text{first bin is empty}) = \frac{(n-1)^m}{n^m} = \left(\frac{n-1}{n}\right)^m \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Uniform Distribution
In discrete mathematics, a uniform distribution means that every outcome in a given set is equally likely.
In our exercise, each ball can be placed into any of the bins with equal probability.
When we say the balls are distributed uniformly at random, it means each bin has the same chance of receiving any specific ball.
This principle is crucial for understanding why we use particular calculations for our problem.
Combinatorial Analysis
Combinatorial analysis involves figuring out the number of ways certain outcomes can occur.
In our problem, combinatorial analysis helps us determine the total number of ways balls can be distributed and the number of ways they can be distributed such that the first bin is empty.
We look at all possible distributions, \( n^m \), and the number of successful distributions where no ball goes into the first bin, \( (n-1)^m \). This is the essence of our favorable outcomes versus total outcomes approach.
Probability Calculation
Calculating probability is about comparing the number of favorable outcomes to the total number of possible outcomes.
For our problem, the total number of ways to distribute m balls into n bins is \( n^m \).
The number of ways to distribute m balls into the remaining \( n-1 \) bins (keeping the first bin empty) is \( (n-1)^m \).
The probability that the first bin remains empty is then the ratio of these values: \[ P(\text{first bin is empty}) = \frac{(n-1)^m}{n^m} = \left( \frac{n-1}{n} \right)^m \].
Binomial Theorem
The binomial theorem is a key concept in probability and combinatorial analysis.
It allows us to expand expressions of the form \( (a + b)^m \) and analyze different combinations of outcomes.
Although not explicitly needed to solve our main problem, understanding the binomial theorem helps deepen our grasp of more complex probability distributions.
The principle behind \( \left( \frac{n-1}{n} \right)^m \) relates inherently to how combinations and probabilities intertwine.

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