This problem requires calculating the likelihood that two boxes have the same number of distinct balls when balls are randomly distributed. The most efficient approach uses state transition dynamic programming to track distributions incrementally while counting valid configurations. This method balances combinatorial explosion with careful memoization to achieve correct probability computation.
Problem Statement
You are given an integer array balls where balls[i] represents the number of balls of the i-th color. The total number of balls is even. All balls are shuffled randomly and then divided equally into two distinct boxes, each containing half of the total balls.
Return the probability that both boxes end up containing the same number of distinct colors. Remember that boxes are distinct, so [color1] in box1 and [color2] in box2 is considered different from [color2] in box1 and [color1] in box2, even if the counts are equal.
Examples
Example 1
Input: balls = [1,1]
Output: 1.00000
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
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
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.
Solution Approach
Use backtracking to explore all distributions
Recursively assign balls of each color to either box while maintaining counts. Track the number of balls and distinct colors in both boxes. Only consider paths where the total number of balls in each box is half of the total. Count valid distributions where the distinct color count is equal.
Apply state transition dynamic programming
Use a DP table indexed by current color index, balls in the first box, and difference in distinct counts. For each color, consider all ways to distribute its balls between the two boxes and update DP states accordingly. This approach avoids recomputation and scales better than naive recursion for multiple colors and balls.
Compute probability from counted configurations
After enumerating all valid distributions, divide the number of configurations with equal distinct counts by the total number of possible distributions. Factorial-based combinatorial formulas handle permutations of identical balls efficiently to get precise probability.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time and space complexity depend on the number of colors k and maximum balls per color. Using DP reduces exponential exploration but requires O(k * n * n) states in the worst case, where n is half the total number of balls.
What Interviewers Usually Probe
- Expect state transition dynamic programming recognition and careful combinatorial reasoning.
- Watch for factorial handling of identical balls to avoid overcounting or undercounting.
- Clarify distinction between boxes early to prevent incorrect probability assumptions.
Common Pitfalls or Variants
Common pitfalls
- Confusing box identity and treating distributions as identical when boxes are distinct.
- Forgetting to include all possible ways to split balls of a single color between boxes.
- Ignoring memoization, leading to exponential runtime for even small input sizes.
Follow-up variants
- Compute probability with more than two boxes while maintaining distinct color equality constraints.
- Calculate probability when some balls are fixed in specific boxes initially, modifying DP states.
- Find expected number of distinct colors in each box instead of probability equality.
How GhostInterview Helps
- GhostInterview guides step-by-step state transitions and keeps track of intermediate distributions efficiently.
- It provides memoization patterns and combinatorial handling to avoid redundant calculations.
- The tool highlights common pitfalls like box distinction errors and factorial miscalculations, ensuring correct probability output.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the key strategy for solving the Probability of Two Boxes Having The Same Number of Distinct Balls?
The key is state transition dynamic programming that tracks distributions of each color and counts of distinct balls while avoiding overcounting permutations.
Can this problem be solved using naive recursion?
Yes, but naive recursion without memoization quickly becomes infeasible due to exponential combinations, making DP essential for larger inputs.
How do I handle identical balls in DP?
Use combinatorial formulas with factorials to compute the number of ways to distribute identical balls between boxes accurately within DP states.
Does the order of boxes matter in probability calculation?
Yes, boxes are distinct, so swapping contents counts as a different distribution. Ignoring this distinction leads to incorrect probability.
What patterns should I recognize for similar problems?
Look for state transition DP patterns with multiple dimensions: current item, subset sums, and differences in counts, often paired with combinatorial counting.
Need direct help with Probability of a Two Boxes Having The Same Number of Distinct Balls instead of spending more time grinding it?
Download GhostInterview when you want a LeetCode solver, not another long practice loop. Capture Probability of a Two Boxes Having The Same Number of Distinct Balls from a screenshot, get the answer path and complexity, and use supported stealth workflows that stay outside captured layers.
Capture the prompt fast instead of rewriting the problem by hand.
Get the solution path, trade-offs, and complexity summary in one pass.
Stay outside captured layers on supported screen-share workflows.
Stay in the same pattern family
Determine the number of ways to reorder an array to get the same binary search tree (BST) from its insertion order.
Open problem page#2597 The Number of Beautiful SubsetsCount all non-empty subsets of an array where no two numbers have an absolute difference equal to k, using array scanning and hash lookup.
Open problem page#1799 Maximize Score After N OperationsMaximize the score after n operations by selecting pairs from the array and using their GCD with dynamic programming or backtracking techniques.
Open problem page