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Jul
5
comment If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?
Actually, an uniform prior means that you made no assumption at all about the fairness of the coin (you still get $p(Head)=0.5$ from ignorance: Since you have no bias, both results are equally likely). A prior 100% assumption that the coin was fair would mean $Pr(p)=\delta(p-0.5)$, and with that you'd find that the coin is fair with probability 1 no matter what you toss. Basically it's the "unshakeable belief" prior. In reality, you'd probably use a prior that's peaked around the fair coin (because most coins are approximately fair, but you wouldn't exclude the probability of it being biased).
Jul
5
comment Which of the following statements are correct?
@DerekHolt: OK, I've looked it up in the Wikipedia, and you're indeed right about the meaning of "order" (I consider that a useless definition since we already have a word for it, cardinality, and a confusing one, since "order" is already used for another notion in group theory).
Jul
5
comment Which of the following statements are correct?
@JonMarkPerry: Wrong in two ways: 1. Nowhere is stated that $x$ or $y$ may not be the identity element. 2. There are groups of order 2 that have more than 2 elements.
Jul
5
comment Which of the following statements are correct?
How do you get $x=y$ in case B?
Jul
4
comment If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?
Actually if you define "fair" as "the probability of landing on head is exactly 0.5" then the probability of the coin being fair is zero, independent of the results you obtained from tossing.
Jul
4
comment Basic encoding with math formula
Also I think the ambiguous sequences are always of the form $01x_101x_2\ldots01x_n01$ vs. $10x_110x_2\ldots10x_n10$ (where the $x_k$ are arbitrary).
Jul
4
comment Basic encoding with math formula
@JonMarkPerry: If $d_n$ is the determinant of the $n\times n$matrix of this form, then we clearly have (by expansion over the first row) $d_n = d_{n-1} - d_{n-2}$. We also easily verify that $d_1=1$ (since that is just the matrix with a single entry $1$) and $d_2=0$ (since that is the matrix with all four elements $1$). So the first few determinants are $1,0,-1,-1,0,1,1,0,\ldots$. One sees that the recursion repeats here. Since a matrix in invertible exactly if its determinant is non-zero, you get a non-invertible matrix exactly if $n=3k-1$.
Jul
4
comment Basic encoding with math formula
Indeed, both $10110$ and $01101$ would be encoded as $12221$. However in general, adding a multiple of $(1,-1,0,1,-1)$ to a valid solution will not give a valid solution, since in a valid solution all entries have to be either $0$ or $1$. Especially if the encoded sequence begins with $2$, you know for sure that the decoded sequence begins with $11$, and therefore the solution is unique.
Jul
4
comment Alternate formulation of Calculus
@GregoryGrant: How do you define piecewise constant functions without defining partitions?
Jul
4
answered Functions $f$ such that $f(z+1)-f(z)$ is holomorphic
Jul
4
comment Functions $f$ such that $f(z+1)-f(z)$ is holomorphic
A concrete example of a nowhere continuous function fulfilling the condition: The product of the Dirichlet functions of the real and the imaginary part. Clearly $f(x+1)=f(x)$, therefore $f(x+1)-f(x)=0$ is entire.
Jul
4
comment Functions $f$ such that $f(z+1)-f(z)$ is holomorphic
Indeed, if $e$ is entire and $p$ is an arbitrary $1$-periodic function, define $f(x)=e(x)+p(x)$, and then $f(x+1)-f(x) = e(x+1)-e(x)$ clearly is entire. However I have no idea if you get all such functions that way.
Jul
2
awarded  Yearling
Jul
1
accepted Decoding the sign expansion of surreal numbers
Jun
28
comment Which is the largest power of natural number that can be evaluated by computers?
@parkhyeyoo: Then just add another disk to increase the storage. However at some point in time we will run out of material to build disks from. However I just suggest to switch to a different representation: Just store numbers as triple (base, mantissa, exponent), and even much larger powers can be stored :-)
Jun
28
comment Which is the largest power of natural number that can be evaluated by computers?
Since there is no need to store the number in RAM, and multi-terabyte disks are common these days, you can even go much farther than that (note that to do multiplications, you only need to have a small part of the number in RAM at any point in time).
Jun
28
comment Which is the largest power of natural number that can be evaluated by computers?
But, in base $10$, a power of $10$ is an 1 followed by a sequence of 0s. So one could argue that the highest power of ten the following code can calculate is limited only by the time until the program is stopped or the computer or the output fails: putchar('1'); while(1) putchar('0');
Jun
27
answered Negative exponents and positive numbers.
Jun
27
revised How is the area of a circle calculated using basic mathematics?
added how to get the correct radius to take for the shell
Jun
27
answered How is the area of a circle calculated using basic mathematics?