ผลต่างระหว่างรุ่นของ "Probstat/notes/balls and bins"
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| แถว 3: | แถว 3: | ||
We consider a balls-and-bins experiment where we throw ''n'' balls independently into ''n'' bins uniformly at random. | We consider a balls-and-bins experiment where we throw ''n'' balls independently into ''n'' bins uniformly at random. | ||
| − | ''How many possible outcomes are there?'' | + | '''Question 1:''' ''How many possible outcomes are there?'' |
Since each ball has ''n'' choices and their choices are independent, there are <math>n^n</math> outcomes. | Since each ball has ''n'' choices and their choices are independent, there are <math>n^n</math> outcomes. | ||
| − | ''What is the probability that bin 1 is empty?'' | + | '''Question 2:''' ''What is the probability that bin 1 is empty?'' |
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| + | In this case, each ball only have ''n'' - 1 choices (because they have to avoid bin 1); therefore there are <math>(n - 1)^n</math> outcomes where bin 1 is empty. Since each outcome is equally likely, the probability that bin 1 is empty is <math>\frac{(n-1)^n}{n^n}=\left(\frac{n-1}{n}\right)^n=\left(1-\frac{1}{n})^n</math>. | ||
รุ่นแก้ไขเมื่อ 03:30, 18 กันยายน 2557
- This is part of probstat. The materials on this part is from this course at Berkeley.
We consider a balls-and-bins experiment where we throw n balls independently into n bins uniformly at random.
Question 1: How many possible outcomes are there?
Since each ball has n choices and their choices are independent, there are outcomes.
Question 2: What is the probability that bin 1 is empty?
In this case, each ball only have n - 1 choices (because they have to avoid bin 1); therefore there are outcomes where bin 1 is empty. Since each outcome is equally likely, the probability that bin 1 is empty is .
