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The probability of a transmission error is 1%. Furthermore, we know that $P(S_1)=p$ and thus $P(S_0)=1-p$. … I know the formulas and I decoded the probability of a transmission error as follows:$$P((S_0\cap E_1)\cup (S_1\cap E_0))$$ I also figured that $P(E_0|S_0)+P(E_1|S_0)=P(E_0|S_1)+P(E_1|S_1)=1$, but I can't …

Consider the following Markov chain on $\{-1, 0, 1\}$. In each step of the Markov chain we flip a coin $Y \in\{0, 1\}$ and a coin $Z\in \{+1, > -1\}$ and set $X_{n+1} = X_n +Y\cdot Z\cdot \mathbf{1}_ …

frac{1}{n}S_n\to -1$ almost surely as $n\to \infty$ Attempt Using Borel-Cantelli, we have $\sum \limits_{n=1}^{\infty}P(X_n=n^{2})=\sum \limits_{n=1}^{\infty}\frac{1}{n^2}<\infty$, which implies the probability … That means $X_n=-1$ for sufficient $n$ with probability $1$. This implies that $$S_n=\frac 1 n \sum\limits_{k=1}^{n} X_k \to -1$$ almost surely. …

Suppose you sit in a class room with $n$ students. There are $k$ question and every student can pick $5$ of them. How many different ways are there such that no question is left out? In total we have …

A bartender and a customer roll a six-sided, ungalvanized cube. If the visitor throws a higher number than the bartender, he wins. A guest is a rascal who cheats. He takes advantage of the waiter's i …

Let $\mathcal{A}$ and $\mathcal{B}$ be $\sigma$ algebras over $\Omega$. I know that $\mathcal{A}\cup \mathcal{B}$ is not a $\sigma$-algebra, but what about $\mathcal{A}\vee \mathcal{B}=\{A\cup B : A\i …

Let $A$, $B$, $C$ be three events with $P[A] = 0.7$, $P[B] = 0.7$ and $P[C] = 0.7$. Let $D$ be the event "exactly one of the three events $A$, $B$, $C$ occurs". What is the maximum value that $P[D]$ c …

If you throw a dice $10$ times, what is the probability that there is at least one $1$ and at least one 6? …

Statement Let $n\in \mathbb{N}$. A càdlàg function $f: [0,T]\to \mathbb{R}$ has only a finite amount of jumps $\Delta f_t=f_t-f_{t-}$ with $|\Delta f_t|\geq 1/n$ Proof (Reductio ad absurdum) Suppose, …

$X$ and $Y$ are defined on $\Omega=\{1,2,3,4\}$ (we have a discrete uniform distribution) where $$X(\omega)=\begin{cases}1, \text{ if } \omega \text{ even} \\ 0, \text{ else}\end{cases} \text{ and }Y( …

I read that as saying "The probability that the event $\{X_n=n^2\}$ will happen infinitely many times is $0$". …

How likely is it if the coin toss is instead in favor of Proposal $B$ with probability $p=0.505$. …

Given $Y$ as a geometrically distributed random variable with $p\in (0,1)$, what is $E[Y|Y\geq 10]?$ I got:$$E(Y|Y\geq 10)=\frac{E(\mathbf{1}_{Y\geq 10}Y)}{P(Y\geq 10)}=\frac{E(\mathbf{1}_{Y\geq 10}Y) …

Given a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_t, P)$ and with notation (regular conditional probabiltiy assumed): $$P^{X|\mathcal{F}_t}(B)=P(X\in B|\mathcal{F}_t)=E(1_B(X)|\ …

Let $Y\sim \operatorname{Geo}(p)$ with $P(Y=k)=p(1-p)^k$. Furthermore, $\hat{Y}\sim \operatorname{NegBin}(2,p)$, i. e. $P(\hat{Y}=k)=kp^2(1-p)^{k-1}$. I want to find $P(\lfloor U\widehat{Y} \rfloor=k) …