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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?
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Suppose that 5 distinct balls are distribution at random into 3 distin...
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.
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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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?
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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? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about 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? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?.
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? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about 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? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?.
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