Probability of a Two Boxes Having The Same Number of Distinct Balls


10 Prerequisite LeetCode Problems
“1467. Probability of a Two Boxes Having The Same Number of Distinct Balls” asks for the probability that after taking some balls from the original box and putting them into another box, both boxes have the same number of distinct balls.
Here are 10 problems to prepare for this one:
70. Climbing Stairs: It’s a basic combinatorics problem. Understanding it will help you with more complex problems involving combinatorics.
357. Count Numbers with Unique Digits: This problem also involves combinatorics and can help you understand how to count permutations.
39. Combination Sum: This problem requires you to find all unique combinations that sum up to a target number, which involves understanding combinatorics and backtracking.
77. Combinations: This problem asks for all combinations for a set of distinct integers. This can help you understand how to generate combinations, which could be useful in the target problem.
526. Beautiful Arrangement: This problem deals with permutations and can help you understand how to navigate permutation problems, which is a key concept in the target problem.
784. Letter Case Permutation: This problem requires generating all possible strings by changing cases. It provides a good exercise in understanding permutations and combinatorics.
46. Permutations: This problem requires you to generate all permutations of a list of unique integers, which will be useful in understanding the mechanics of permutation and combinatorics.
62. Unique Paths: This problem is a basic dynamic programming problem, understanding it would help you with more complex problems involving combinatorics and probability.
494. Target Sum: This problem involves finding all possible ways to assign symbols to make the sum of numbers equal to target S, which involves understanding combinatorics and backtracking.
688. Knight Probability in Chessboard: This problem asks you to calculate the probability that the knight remains on the board after it has moved a certain number of times. This problem can help you understand how to calculate probabilities in combinatoric problems.
“Probability of a Two Boxes Having The Same Number of Distinct Balls” involves combinatorics, probability, and recursion. Here are ten simpler problems to get ready for this one:
Unique Binary Search Trees: This is a good starter problem to understand combinatorics.
Generate Parentheses: This problem helps in understanding recursive solution formulation.
Subsets: This problem helps to understand the recursive generation of combinations.
Partition Equal Subset Sum: This problem is a good practice for understanding the concept of partitioning and recursion.
These cover combinatorics, probability, and recursion, which are all necessary to solve the “Probability of a Two Boxes Having The Same Number of Distinct Balls” problem.
Problem Classification
Problem Statement: Given 2n balls of k distinct colors. You will be given an integer array balls of size k where balls[i] is the number of balls of color i.
All the balls will be shuffled uniformly at random, then we will distribute the first n balls to the first box and the remaining n balls to the other box (Please read the explanation of the second example carefully).
Please note that the two boxes are considered different. For example, if we have two balls of colors a and b, and two boxes [] and (), then the distribution [a] (b) is considered different than the distribution [b] (a) (Please read the explanation of the first example carefully).
Return the probability that the two boxes have the same number of distinct balls. Answers within 105 of the actual value will be accepted as correct.
Example 1:
Input: balls = [1,1] Output: 1.00000 Explanation: Only 2 ways to divide the balls equally:
 A ball of color 1 to box 1 and a ball of color 2 to box 2
 A ball of color 2 to box 1 and a ball of color 1 to box 2 In both ways, the number of distinct colors in each box is equal. The probability is 2/2 = 1
Example 2:
Input: balls = [2,1,1] Output: 0.66667 Explanation: We have the set of balls [1, 1, 2, 3] This set of balls will be shuffled randomly and we may have one of the 12 distinct shuffles with equal probability (i.e. 1/12): [1,1 / 2,3], [1,1 / 3,2], [1,2 / 1,3], [1,2 / 3,1], [1,3 / 1,2], [1,3 / 2,1], [2,1 / 1,3], [2,1 / 3,1], [2,3 / 1,1], [3,1 / 1,2], [3,1 / 2,1], [3,2 / 1,1] After that, we add the first two balls to the first box and the second two balls to the second box. We can see that 8 of these 12 possible random distributions have the same number of distinct colors of balls in each box. Probability is 8/12 = 0.66667
Example 3:
Input: balls = [1,2,1,2] Output: 0.60000 Explanation: The set of balls is [1, 2, 2, 3, 4, 4]. It is hard to display all the 180 possible random shuffles of this set but it is easy to check that 108 of them will have the same number of distinct colors in each box. Probability = 108 / 180 = 0.6
Constraints:
1 <= balls.length <= 8 1 <= balls[i] <= 6 sum(balls) is even.
Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.
Distilling the Problem to Its Core Elements
In order to have me distill a problem to its core, you could ask questions that prompt for a deeper analysis of the problem, understanding of the underlying concepts, and simplification of the problem’s essence. Here are some examples of such prompts:
 Can you identify the fundamental concept or principle this problem is based upon? Please explain.
 What is the simplest way you would describe this problem to someone unfamiliar with the subject?
 What is the core problem we are trying to solve? Can we simplify the problem statement?
 Can you break down the problem into its key components?
 What is the minimal set of operations we need to perform to solve this problem?
These prompts guide the discussion towards simplifying the problem, stripping it down to its essential elements, and understanding the core problem to be solved.
Visual Model of the Problem
How to visualize the problem statement for this problem?
Problem Restatement
Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.
Abstract Representation of the Problem
Could you help me formulate an abstract representation of this problem?
Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific realworld details?
Terminology
Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?
Problem Simplification and Explanation
Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?
Constraints
Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.
What are the key insights from analyzing the constraints?
Case Analysis
Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.
Provide names by categorizing these cases
What are the key insights from analyzing the different cases?
Identification of Applicable Theoretical Concepts
Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?
Problem Breakdown and Solution Methodology
Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.
Inference of ProblemSolving Approach from the Problem Statement
How did you infer from the problem statement that this problem can be solved using ?
Stepwise Refinement
Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the highlevel solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?
Solution Approach and Analysis
Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.
Thought Process
Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.
From Brute Force to Optimal Solution
Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?
Code Explanation and Design Decisions
Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?
Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.
Coding Constructs
Consider the following piece of complex software code.
What are the highlevel problemsolving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a nonprogrammer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?
Language Agnostic Coding Drills
Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.
Dissect the code and identify each distinct concept it contains. Remember, this process should be languageagnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problemsolving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the stepbystep process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.
Targeted Drills in Python
Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Pythonbased coding drills for each of those concepts.
Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problemspecific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.
Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.
Q&A
Similar Problems
Can you suggest 10 problems from LeetCode that require similar problemsolving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem.