Linked Questions

55
votes
26answers
5k views

Applications of complex numbers to solve non-complex problems

Recently I asked a question regarding the diophantine equation $x^2+y^2=z^n$ for $x, y, z, n \in \mathbb{N}$, which to my surprise was answered with the help complex numbers. I find it fascinating ...
54
votes
5answers
17k views

Probability for the length of the longest run in $n$ Bernoulli trials

Suppose a biased coin (probability of head being $p$) was flipped $n$ times. I would like to find the probability that the length of the longest run of heads, say $\ell_n$, exceeds a given number $m$, ...
25
votes
3answers
13k views

Expectation of the maximum of i.i.d. geometric random variables

Given $n$ independent geometric random variables $X_n$, each with probability parameter $p$ (and thus expectation $E\left(X_n\right) = \frac{1}{p}$), what is $$E_n = E\left(\max_{i \in 1 .. n}X_n\...
31
votes
5answers
1k views

Unexpected Proofs Using Generating Functions

I recently came across this beautiful proof by Erdős that uses generating functions in a unique way: Let $S = \{a_1, \cdots, a_n \}$ be a finite set of positive integers such that no two subsets of ...
1
vote
4answers
4k views

Asymptotic behaviour of the logarithm

In this post, the poster suspected that the $\log$ function would eventually flatten out and approach a straight line. We all know this isn't true of course. But then a commenter pointed out this: @...
2
votes
1answer
712 views

If I flip a coin $n$ times, what is the expected maximum number of heads or tails in a row?

Question: If I flip a coin $n$ times, what is the maximum number of heads or tails in a row that I should expect?
14
votes
1answer
369 views

On 5-adic representation of square root of -1

Let $\alpha \in \mathbb{Z}_5$ be the solution to $f(x):=x^2+1=0$ such that $\alpha \equiv 2 \, (\text{mod} \,5)$ (that we obtain by Hensel's lemma). Then $$ \alpha = \sum_{k=0}^{\infty} a_k \, 5^k$$ ...
4
votes
1answer
315 views

Probability of Longest Head Run when $p\rightarrow 1$

Let $L_n$ be the largest contiguous heads sequence in $n$ coin tosses with $p$ probability of having head. It is known that $$ \forall \epsilon>0 \lim\limits_{n\to\infty}\mathbb{P} \left(\left|\...
0
votes
0answers
38 views

is it possible to find out how many times coin will return head in n-sequence?

I would like to ask: If we throw a coin 1000 times, expected result is we will get 500 heads from it, correct? Now, is it possible to calculate how many times (max value) head will be selected in ...
0
votes
0answers
26 views

Does the number of the same side appearing in a row converge to a finite number? Fair coin toss.

Toss a fair coin N times. Does the number of the same side appearing in a row converge to a finite number? If yes, what is it in terms of N? Are confidence intervals needed? Thank you!