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Round trip stones [FINALIZED]
#3: Post edited
Round trip stones
- Round trip stones [FINALIZED]
- $N$ vessels initially contain $3$ stones each. What is the probability of having at least $3$ stones in the first vessel after moving a uniformly random selection from the first vessel to the second, then from the second vessel to the third, and so on back to the first vessel?
- ## Input
- - A positive integer $N$.
- - Your code must work for inputs up to and including $10$, but may crash, error, or give incorrect output for larger inputs.
- ## Output
- - A probability $p$, so $0 \le p \le 1$.
- - Your output is valid if rounding it to $6$ decimal places results in the output shown in the relevant test case.
- - This is the probability of the first vessel containing greater than or equal to $3$ stones after the following process:
- - Start with $N$ vessels containing $3$ stones each.
- - For each vessel from $1$ to $N$ in order:
- - Choose uniformly randomly between $0$ and all of the stones from the vessel, and move them to the next vessel.
- - Note that when moving from vessel $N$, they are moved back to vessel $1$.
- ## Examples
- ### Input 1
- When there is only $1$ vessel, all of the stones moved from it are moved back into it, resulting in exactly $3$ stones in vessel $1$, so the probability of there being at least $3$ stones in vessel $1$ is $1$.
- ### Input 2
- When there are $2$ vessels, the process goes like this:
- - $0$, $1$, $2$, or $3$ stones are taken from vessel $1$ and added to vessel $2$. For example, $2$ stones may be taken from vessel $1$ and added to vessel $2$.
- - There are now $5$ stones in vessel $2$, so $0$, $1$, $2$, $3$, $4$, or $5$ stones are taken from vessel $2$ and moved back to vessel $1$.
- There are $22$ ways that this process can happen, and $16$ of those ways result in $3$ or more stones in vessel $1$. This means that the probability of there being at least $3$ stones in vessel $1$ is $16/22=0.727273$ (rounded to $6$ decimal places).
- ### Input 3
- When there are $3$ vessels, the process goes like this:
- - $0$, $1$, $2$, or $3$ stones are taken from vessel $1$ and added to vessel $2$. For example, $2$ stones may be taken from vessel $1$ and added to vessel $2$.
- - There are now $5$ stones in vessel $2$, so $0$, $1$, $2$, $3$, $4$, or $5$ stones are taken from vessel $2$ and added to vessel $3$. For example, $5$ stones may be taken from vessel $2$ and added to vessel $3$.
- - There are now $8$ stones in vessel $3$, so $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, or $8$ stones are taken from vessel $3$ and moved back to vessel $1$.
- There are $140$ ways that this process can happen, and $102$ of those ways result in $3$ or more stones in vessel $1$. The probability of there being at least $3$ stones in vessel $1$ is $102/140=0.728571$ (rounded to $6$ decimal places).
- ## Test cases
- Test cases are in the format `input : output`, and are rounded to 6 decimal places. For example, an output of `0.727272727` is equivalent to an output of `0.727273` when rounded.
- ```text
- 1 : 1.000000
- 2 : 0.727273
- 3 : 0.728571
- 4 : 0.736842
- 5 : 0.743789
- 6 : 0.749164
- 7 : 0.753344
- 8 : 0.756657
- 9 : 0.759336
- 10 : 0.761542
- ```
- ## Scoring
- This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
- > Explanations are optional, but I'm more likely to upvote answers that have one.
- [code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
- ## Now posted: [Round trip stones](https://codegolf.codidact.com/posts/291293)
- ---
- $N$ vessels initially contain $3$ stones each. What is the probability of having at least $3$ stones in the first vessel after moving a uniformly random selection from the first vessel to the second, then from the second vessel to the third, and so on back to the first vessel?
- ## Input
- - A positive integer $N$.
- - Your code must work for inputs up to and including $10$, but may crash, error, or give incorrect output for larger inputs.
- ## Output
- - A probability $p$, so $0 \le p \le 1$.
- - Your output is valid if rounding it to $6$ decimal places results in the output shown in the relevant test case.
- - This is the probability of the first vessel containing greater than or equal to $3$ stones after the following process:
- - Start with $N$ vessels containing $3$ stones each.
- - For each vessel from $1$ to $N$ in order:
- - Choose uniformly randomly between $0$ and all of the stones from the vessel, and move them to the next vessel.
- - Note that when moving from vessel $N$, they are moved back to vessel $1$.
- ## Examples
- ### Input 1
- When there is only $1$ vessel, all of the stones moved from it are moved back into it, resulting in exactly $3$ stones in vessel $1$, so the probability of there being at least $3$ stones in vessel $1$ is $1$.
- ### Input 2
- When there are $2$ vessels, the process goes like this:
- - $0$, $1$, $2$, or $3$ stones are taken from vessel $1$ and added to vessel $2$. For example, $2$ stones may be taken from vessel $1$ and added to vessel $2$.
- - There are now $5$ stones in vessel $2$, so $0$, $1$, $2$, $3$, $4$, or $5$ stones are taken from vessel $2$ and moved back to vessel $1$.
- There are $22$ ways that this process can happen, and $16$ of those ways result in $3$ or more stones in vessel $1$. This means that the probability of there being at least $3$ stones in vessel $1$ is $16/22=0.727273$ (rounded to $6$ decimal places).
- ### Input 3
- When there are $3$ vessels, the process goes like this:
- - $0$, $1$, $2$, or $3$ stones are taken from vessel $1$ and added to vessel $2$. For example, $2$ stones may be taken from vessel $1$ and added to vessel $2$.
- - There are now $5$ stones in vessel $2$, so $0$, $1$, $2$, $3$, $4$, or $5$ stones are taken from vessel $2$ and added to vessel $3$. For example, $5$ stones may be taken from vessel $2$ and added to vessel $3$.
- - There are now $8$ stones in vessel $3$, so $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, or $8$ stones are taken from vessel $3$ and moved back to vessel $1$.
- There are $140$ ways that this process can happen, and $102$ of those ways result in $3$ or more stones in vessel $1$. The probability of there being at least $3$ stones in vessel $1$ is $102/140=0.728571$ (rounded to $6$ decimal places).
- ## Test cases
- Test cases are in the format `input : output`, and are rounded to 6 decimal places. For example, an output of `0.727272727` is equivalent to an output of `0.727273` when rounded.
- ```text
- 1 : 1.000000
- 2 : 0.727273
- 3 : 0.728571
- 4 : 0.736842
- 5 : 0.743789
- 6 : 0.749164
- 7 : 0.753344
- 8 : 0.756657
- 9 : 0.759336
- 10 : 0.761542
- ```
- ## Scoring
- This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
- > Explanations are optional, but I'm more likely to upvote answers that have one.
- [code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
#2: Post edited
- $N$ vessels initially contain $3$ stones each. What is the probability of having at least $3$ stones in the first vessel after moving a uniformly random selection from the first vessel to the second, then from the second vessel to the third, and so on back to the first vessel?
- ## Input
- - A positive integer $N$.
- - Your code must work for inputs up to and including $10$, but may crash, error, or give incorrect output for larger inputs.
- ## Output
- - A probability $p$, so $0 \le p \le 1$.
- - Your output is valid if rounding it to $6$ decimal places results in the output shown in the relevant test case.
- - This is the probability of the first vessel containing greater than or equal to $3$ stones after the following process:
- - Start with $N$ vessels containing $3$ stones each.
- - For each vessel from $1$ to $N$ in order:
- - Choose uniformly randomly between $0$ and all of the stones from the vessel, and move them to the next vessel.
- - Note that when moving from vessel $N$, they are moved back to vessel $1$.
- ## Examples
- ### Input 1
- When there is only $1$ vessel, all of the stones moved from it are moved back into it, resulting in exactly $3$ stones in vessel $1$, so the probability of there being at least $3$ stones in vessel $1$ is $1$.
- ### Input 2
- When there are $2$ vessels, the process goes like this:
- - $0$, $1$, $2$, or $3$ stones are taken from vessel $1$ and added to vessel $2$. For example, $2$ stones may be taken from vessel $1$ and added to vessel $2$.
- - There are now $5$ stones in vessel $2$, so $0$, $1$, $2$, $3$, $4$, or $5$ stones are taken from vessel $2$ and moved back to vessel $1$.
There are $22$ ways that this process can happen, and $16$ of those ways result in $3$ or more stones in vessel $1$. This means the probability of there being at least $3$ stones in vessel $1$ is $16/22=0.727273$ (rounded to $6$ decimal places).- ### Input 3
- When there are $3$ vessels, the process goes like this:
- - $0$, $1$, $2$, or $3$ stones are taken from vessel $1$ and added to vessel $2$. For example, $2$ stones may be taken from vessel $1$ and added to vessel $2$.
- - There are now $5$ stones in vessel $2$, so $0$, $1$, $2$, $3$, $4$, or $5$ stones are taken from vessel $2$ and added to vessel $3$. For example, $5$ stones may be taken from vessel $2$ and added to vessel $3$.
- - There are now $8$ stones in vessel $3$, so $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, or $8$ stones are taken from vessel $3$ and moved back to vessel $1$.
- There are $140$ ways that this process can happen, and $102$ of those ways result in $3$ or more stones in vessel $1$. The probability of there being at least $3$ stones in vessel $1$ is $102/140=0.728571$ (rounded to $6$ decimal places).
- ## Test cases
- Test cases are in the format `input : output`, and are rounded to 6 decimal places. For example, an output of `0.727272727` is equivalent to an output of `0.727273` when rounded.
- ```text
- 1 : 1.000000
- 2 : 0.727273
- 3 : 0.728571
- 4 : 0.736842
- 5 : 0.743789
- 6 : 0.749164
- 7 : 0.753344
- 8 : 0.756657
- 9 : 0.759336
- 10 : 0.761542
- ```
- ## Scoring
- This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
- > Explanations are optional, but I'm more likely to upvote answers that have one.
- [code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
- $N$ vessels initially contain $3$ stones each. What is the probability of having at least $3$ stones in the first vessel after moving a uniformly random selection from the first vessel to the second, then from the second vessel to the third, and so on back to the first vessel?
- ## Input
- - A positive integer $N$.
- - Your code must work for inputs up to and including $10$, but may crash, error, or give incorrect output for larger inputs.
- ## Output
- - A probability $p$, so $0 \le p \le 1$.
- - Your output is valid if rounding it to $6$ decimal places results in the output shown in the relevant test case.
- - This is the probability of the first vessel containing greater than or equal to $3$ stones after the following process:
- - Start with $N$ vessels containing $3$ stones each.
- - For each vessel from $1$ to $N$ in order:
- - Choose uniformly randomly between $0$ and all of the stones from the vessel, and move them to the next vessel.
- - Note that when moving from vessel $N$, they are moved back to vessel $1$.
- ## Examples
- ### Input 1
- When there is only $1$ vessel, all of the stones moved from it are moved back into it, resulting in exactly $3$ stones in vessel $1$, so the probability of there being at least $3$ stones in vessel $1$ is $1$.
- ### Input 2
- When there are $2$ vessels, the process goes like this:
- - $0$, $1$, $2$, or $3$ stones are taken from vessel $1$ and added to vessel $2$. For example, $2$ stones may be taken from vessel $1$ and added to vessel $2$.
- - There are now $5$ stones in vessel $2$, so $0$, $1$, $2$, $3$, $4$, or $5$ stones are taken from vessel $2$ and moved back to vessel $1$.
- There are $22$ ways that this process can happen, and $16$ of those ways result in $3$ or more stones in vessel $1$. This means that the probability of there being at least $3$ stones in vessel $1$ is $16/22=0.727273$ (rounded to $6$ decimal places).
- ### Input 3
- When there are $3$ vessels, the process goes like this:
- - $0$, $1$, $2$, or $3$ stones are taken from vessel $1$ and added to vessel $2$. For example, $2$ stones may be taken from vessel $1$ and added to vessel $2$.
- - There are now $5$ stones in vessel $2$, so $0$, $1$, $2$, $3$, $4$, or $5$ stones are taken from vessel $2$ and added to vessel $3$. For example, $5$ stones may be taken from vessel $2$ and added to vessel $3$.
- - There are now $8$ stones in vessel $3$, so $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, or $8$ stones are taken from vessel $3$ and moved back to vessel $1$.
- There are $140$ ways that this process can happen, and $102$ of those ways result in $3$ or more stones in vessel $1$. The probability of there being at least $3$ stones in vessel $1$ is $102/140=0.728571$ (rounded to $6$ decimal places).
- ## Test cases
- Test cases are in the format `input : output`, and are rounded to 6 decimal places. For example, an output of `0.727272727` is equivalent to an output of `0.727273` when rounded.
- ```text
- 1 : 1.000000
- 2 : 0.727273
- 3 : 0.728571
- 4 : 0.736842
- 5 : 0.743789
- 6 : 0.749164
- 7 : 0.753344
- 8 : 0.756657
- 9 : 0.759336
- 10 : 0.761542
- ```
- ## Scoring
- This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
- > Explanations are optional, but I'm more likely to upvote answers that have one.
- [code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
#1: Initial revision
Round trip stones
$N$ vessels initially contain $3$ stones each. What is the probability of having at least $3$ stones in the first vessel after moving a uniformly random selection from the first vessel to the second, then from the second vessel to the third, and so on back to the first vessel? ## Input - A positive integer $N$. - Your code must work for inputs up to and including $10$, but may crash, error, or give incorrect output for larger inputs. ## Output - A probability $p$, so $0 \le p \le 1$. - Your output is valid if rounding it to $6$ decimal places results in the output shown in the relevant test case. - This is the probability of the first vessel containing greater than or equal to $3$ stones after the following process: - Start with $N$ vessels containing $3$ stones each. - For each vessel from $1$ to $N$ in order: - Choose uniformly randomly between $0$ and all of the stones from the vessel, and move them to the next vessel. - Note that when moving from vessel $N$, they are moved back to vessel $1$. ## Examples ### Input 1 When there is only $1$ vessel, all of the stones moved from it are moved back into it, resulting in exactly $3$ stones in vessel $1$, so the probability of there being at least $3$ stones in vessel $1$ is $1$. ### Input 2 When there are $2$ vessels, the process goes like this: - $0$, $1$, $2$, or $3$ stones are taken from vessel $1$ and added to vessel $2$. For example, $2$ stones may be taken from vessel $1$ and added to vessel $2$. - There are now $5$ stones in vessel $2$, so $0$, $1$, $2$, $3$, $4$, or $5$ stones are taken from vessel $2$ and moved back to vessel $1$. There are $22$ ways that this process can happen, and $16$ of those ways result in $3$ or more stones in vessel $1$. This means the probability of there being at least $3$ stones in vessel $1$ is $16/22=0.727273$ (rounded to $6$ decimal places). ### Input 3 When there are $3$ vessels, the process goes like this: - $0$, $1$, $2$, or $3$ stones are taken from vessel $1$ and added to vessel $2$. For example, $2$ stones may be taken from vessel $1$ and added to vessel $2$. - There are now $5$ stones in vessel $2$, so $0$, $1$, $2$, $3$, $4$, or $5$ stones are taken from vessel $2$ and added to vessel $3$. For example, $5$ stones may be taken from vessel $2$ and added to vessel $3$. - There are now $8$ stones in vessel $3$, so $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, or $8$ stones are taken from vessel $3$ and moved back to vessel $1$. There are $140$ ways that this process can happen, and $102$ of those ways result in $3$ or more stones in vessel $1$. The probability of there being at least $3$ stones in vessel $1$ is $102/140=0.728571$ (rounded to $6$ decimal places). ## Test cases Test cases are in the format `input : output`, and are rounded to 6 decimal places. For example, an output of `0.727272727` is equivalent to an output of `0.727273` when rounded. ```text 1 : 1.000000 2 : 0.727273 3 : 0.728571 4 : 0.736842 5 : 0.743789 6 : 0.749164 7 : 0.753344 8 : 0.756657 9 : 0.759336 10 : 0.761542 ``` ## Scoring This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins. > Explanations are optional, but I'm more likely to upvote answers that have one. [code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"