Found problems: 70
Seven points are selected randomly from $ S^1\subset\mathbb C$. What is the probability that origin is not contained in convex hull of these points?
Let $X$ and $Y$ be independent, identically distributed random points on the unit sphere in $\mathbb{R}^3$. For which distribution of $X$ will the expectation of the (Euclidean) distance of $X$ and $Y$ be maximal?
In a lake there are several sorts of fish, in the following distribution: $ 18\%$ catfish, $ 2\%$ sturgeon and $ 80\%$ other. Of a catch of ten fishes, let $ x$ denote the number of the catfish and $ y$ that of the sturgeons. Find the expectation of $ \frac {x}{y \plus{} 1}$
Cast a dice $n$ times. Denote by $X_1,\ X_2,\ \cdots ,\ X_n$ the numbers shown on each dice. Define $Y_1,\ Y_2,\ \cdots,\ Y_n$ by
\[Y_1=X_1,\ Y_k=X_k+\frac{1}{Y_{k-1}}\ (k=2,\ \cdots,\ n)\]
Find the probability $p_n$ such that $\frac{1+\sqrt{3}}{2}\leq Y_n\leq 1+\sqrt{3}.$
35 points
Suppose that there are 100 seats in a saloon for 100 students. All students except one know their seat. First student (which is the one who doesn't know his seat) comes to the saloon and sits randomly somewhere. Then others enter the saloon one by one. Every student that enters the saloon and finds his seat vacant, sits there and if he finds his seat occupied he sits somewhere else randomly. Find the probability that last two students sit on their seats.
A coastal battery sights an enemy cruiser lying one kilometer off the coast and opens fire on it at the rate of one round per minute. After the first shot, the cruiser begins to move away at a speed of $ 60$ kilometers an hour. Let the probability of a hit be $ 0.75x^{ \minus{} 2}$, where $ x$ denotes the distance (in kilometers) between the cruiser and the coast ($ x\geq 1$), and suppose that the battery goes on firing till the cruiser either sinks or disappears. Further, let the probability of the cruiser sinking after $ n$ hits be $ 1 \minus{} \frac {1}{4^n}$ ($ n \equal{} 0,1,...$). Show that the probability of the cruiser escaping is $ \frac {2\sqrt {2}}{3\pi}$
Prove that we can put $ \Omega(\frac1{\epsilon})$ points on surface of a sphere with radius 1 such that distance of each of these points and the plane passing through center and two of other points is at least $ \epsilon$.
We roll a regular 6-sided dice $n$ times. What is the probabilty that the total number of eyes rolled is a multiple of 5?
In a contest there are $ n$ yes-no problems. We know that no two contestants have the same set of answers. To each question we give a random uniform grade of set $ \{1,2,3,\dots,2n\}$. Prove that the probability that exactly one person gets first is at least $ \frac12$.
Let $n$ and $k$ be fixed positive integers , and $a$ be arbitrary nonnegative integer .
Choose a random $k$-element subset $X$ of $\{1,2,...,k+a\}$ uniformly (i.e., all k-element subsets are chosen with the same probability) and, independently of $X$, choose random n-elements subset $Y$ of $\{1,2,..,k+a+n\}$ uniformly.
Prove that the probability
$P\left( \text{min}(Y)>\text{max}(X)\right)$
does not depend on $a$.
a government has decided to help it's people by giving them $n$ coupons for $n$ fundamental things, but because of being unmanaged, the giving of the coupons to the people is random. in each time that a person goes to the office to get a coupon, the office manager gives him one of the $n$ coupons randomly and with the same probability. It's obvious that in this system a person may get a coupon that he had it before.
suppose that $X_n$ is the random varieble of the first time that a person gets all of the $n$ coupons. show that $\frac{X_n}{n ln(n)}$ in probability converges to $1$.
Let $X_0, \xi_{i, j}, \epsilon_k$ (i, j, k ∈ N) be independent, non-negative integer random variables. Suppose that $\xi_{i, j}$ (i, j ∈ N) have the same distribution, $\epsilon_k$ (k ∈ N) also have the same distribution.
$\mathbb{E}(\xi_{1,1})=1$ , $\mathbb{E}(X_0^l)<\infty$ , $\mathbb{E}(\xi_{1,1}^l)<\infty$ , $\mathbb{E}(\epsilon_1^l)<\infty$ for some $l\in\mathbb{N}$
Consider the random variable $X_n := \epsilon_n + \sum_{j=1}^{X_{n-1}} \xi_{n,j}$ (n ∈ N) , where $\sum_{j=1}^0 \xi_{n,j} :=0$
Introduce the sequence $M_n := X_n-X_{n-1}-\mathbb{E}(\epsilon_n)$ (n ∈ N)
Prove that there is a polynomial P of degree $\leq l/2$ such that $\mathbb{E}(M_n^l) = P_l(n)$ (n ∈ N).
Throw a die $n$ times, let $a_k$ be a number shown on the die in the $k$-th place. Define $s_n$ by $s_n=\sum_{k=1}^n 10^{n-k}a_k$.
(1) Find the probability such that $s_n$ is divisible by 4.
(2) Find the probability such that $s_n$ is divisible by 6.
(3) Find the probability such that $s_n$ is divisible by 7.
Last Edited
Thanks, jmerry & JBL
A permutation $ \pi$ is selected randomly through all $ n$-permutations.
a) if \[ C_a(\pi)\equal{}\mbox{the number of cycles of length }a\mbox{ in }\pi\] then prove that $ E(C_a(\pi))\equal{}\frac1a$
b) Prove that if $ \{a_1,a_2,\dots,a_k\}\subset\{1,2,\dots,n\}$ the probability that $ \pi$ does not have any cycle with lengths $ a_1,\dots,a_k$ is at most $ \frac1{\sum_{i\equal{}1}^ka_i}$
At first a fair dice is placed in such way the spot $1$ is shown on the top face. After that, select a face from the four sides at random, the process that the side would be the top face is repeated $n$ times. Note the sum of the spots of opposite face is 7.
(1) Find the probability such that the spot $1$ would appear on the top face after the $n$-repetition.
(2) Find the expected value of the number of the spot on the top face after the $n$-repetition.
Let $ X_1,X_2,\ldots, X_n$ be independent, identically distributed, nonnegative random variables with a common continuous distribution function $ F$. Suppose in addition that the inverse of $ F$, the quantile function $ Q$, is also continuous and $ Q(0)=0$. Let $ 0=X_{0: n} \leq X_{1: n} \leq \ldots \leq X_{n: n}$ be the ordered sample from the above random variables. Prove that if $ EX_1$ is finite, then the random variable \[ \Delta = \sup_{0\leq y \leq 1} \left| \frac 1n \sum_{i=1}^{\lfloor ny \rfloor +1} (n+1-i)(X_{i: n}-X_{i-1: n})- \int_0^y (1-u)dQ(u) \right|\] tends to zero with probability one as $ n \rightarrow \infty$.
[i]S. Csorgp, L. Horvath[/i]
For $[0,1]\subset E\subset [0,+\infty)$ where $E$ is composed of a finite number of closed interval,we start a two dimensional Brownian motion from the point $x<0$ terminating when we first hit $E$.Let $p(x)$ be the probability of the finishing point being in $[0,1]$.Prove that $p(x)$ is increasing on $[-1,0)$.
Is there a gambling game with an honest coin for two players, in which the probability of one of them winning is $\frac{1}{{\pi}^e}$.
Let F(x) be a known distribution function, the random variables $\eta_1 , \eta_2 ...$ be independent of the common distribution function $F( x - \vartheta)$, where $\vartheta$ is the shift parameter. Let us call the shift parameter "well estimated" if there exists a positive constant c, so that any of $\varepsilon> 0$ there exist a Lebesgue measure $\varepsilon$ Borel set E ("confidence set") and a Borel-measurable function $t_n( x_1 ,. .., x_n )$ ( n = 1,2, ...) such that for any $\vartheta$ we have
$$P ( \vartheta- t_n ( \eta_1 , ..., \eta_n ) \in E )> 1-e^{-cn} \qquad( n > n_0 ( \varepsilon, F ) )$$
Prove that
a) if F is not absolutely continuous, then the shift parameter is "well estimated",
b) if F is absolutely continuous and F' is continuous, then it is not "well estimated".
Let $ p_0,p_1,\ldots$ be a probability distribution on the set of nonnegative integers. Select a number according to this distribution and repeat the selection independently until either a zero or an already selected number is obtained. Write the selected numbers in a row in order of selection without the last one. Below this line, write the numbers again in increasing order. Let $ A_i$ denote the event that the number $ i$ has been selected and that it is in the same place in both lines. Prove that the events $ A_i \;(i\equal{}1,2,\ldots)$ are mutually independent, and $ P(A_i)\equal{}p_i$.
[i]T. F. Mori[/i]
A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled and n cards are dealt to $A$ and $B$. $A$ and $B$ alternately discard a card face up, starting with $A$. The game when the sum of the discards is first divisible by $2n + 1$, and the last person to discard wins. What is the probability that $A$ wins if neither player makes a mistake?
Let $ X_1,X_2,...,X_n$ be (not necessary independent) discrete random variables. Prove that there exist at least $ n^2/2$ pairs
$ (i,j)$ such that \[ H(X_i\plus{}X_j) \geq \frac 13 \min_{1 \leq k \leq n} \{ H(X_k) \},\] where $ H(X)$ denotes the Shannon entropy of $ X$.
[i]GY. Katona[/i]
Sarah and Darah play the following game. Sarah puts $ n$ coins numbered with $ 1,\dots,n$ on a table (Each coin is in HEAD or TAIL position.) At each step Darah gives a coin to Sarah and she (Sarah) let him (Dara) to change the position of all coins with number multiple of a desired number $ k$. At the end, all of the coins that are in TAIL position will be given to Sarah and all of the coins with HEAD position will be given to Darah. Prove that Sarah can put the coins in a position at the beginning of the game such that she gains at least $ \Omega(n)$ coins.
[hide="Hint:"]Chernov inequality![/hide]
Consider a computer network consisting of servers and bi-directional communication channels among them. Unfortunately, not all channels operate. Each direction of each channel fails with probability $p$ and operates otherwise. (All of these stochastic events are mutually independent, and $0 \le p \le 1$.) There is a root serve, denoted by $r$. We call the network [i]operational[/i], if all serves can reach $r$ using only operating channels. Note that we do not require $r$ to be able to reach any servers.
Show that the probability of the network to be operational does not depend on the choice of $r$. (In other words, for any two distinct root servers $r_1$ and $r_2$, the operational probability is the same.)
A gambler plays the following coin-tossing game. He can bet an arbitrary positive amount of money. Then a fair coin is tossed, and the gambler wins or loses the amount he bet depending on the outcome. Our gambler, who starts playing with $ x$ forints, where $ 0<x<2C$, uses the following strategy: if at a given time his capital is $ y<C$, he risks all of it; and if he has $ y>C$, he only bets $ 2C\minus{}y$. If he has exactly $ 2C$ forints, he stops playing. Let $ f(x)$ be the probability that he reaches $ 2C$ (before going bankrupt). Determine the value of $ f(x)$.