Suppose that 5 distinct balls are distribution at random into 3 distinct boxes in such a way that each of the 5 balls can get into any are of the 3 boxes. Find the probability that exactly one box is empty?

Question

Answer

Understanding the Problem
To find the probability that exactly one box is empty when distributing 5 distinct balls into 3 distinct boxes, we first need to calculate the total number of distributions and the favorable outcomes.
Total Distributions
- Each of the 5 balls can go into any of the 3 boxes.
- Therefore, the total number of ways to distribute the balls is 3^5.
Favorable Outcomes for Exactly One Box Empty
1. Choose the Empty Box:
- There are 3 ways to choose which box will be empty.
2. Distributing Balls:
- The remaining 2 boxes must contain all 5 balls.
- Each ball can go into either of the 2 boxes. Hence, there are 2^5 ways to distribute the balls among the 2 boxes.
3. Calculate Favorable Outcomes:
- Total favorable outcomes = Number of ways to choose the empty box * Number of ways to distribute the balls
- This gives us: 3 * 2^5.
Calculating the Probability
- Total distributions = 3^5 = 243.
- Favorable outcomes = 3 * 32 = 96.
- Therefore, the probability that exactly one box is empty can be calculated as follows:
Probability = (Number of favorable outcomes) / (Total distributions)
Probability = 96 / 243.
Final Result
- The probability that exactly one box is empty when distributing 5 distinct balls into 3 distinct boxes is 96/243.

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