We have thrown $k$ white dice and $m$ black dice. Find the probability that the remainder modulo $7$ of the sum of the numbers on the white dice is equal to the remainder modulo $7$ of the sum of the numbers on the black dice.

You have to organize a fair procedure to randomly select someone from $ n$ people so that every one of them would be chosen with the probability $ \frac{1}{n}$. You are allowed to choose two real numbers $ 0<p_1<1$ and $ 0<p_2<1$ and order two coins which satisfy the following requirement: the probability of tossing "heads" on the first coin $ p_1$ and the probability of tossing "heads" on the second coin is $ p_2$. Before starting the procedure, you are supposed to announce an upper bound on the total number of times that the two coins are going to be flipped altogether. Describe a procedure that achieves this goal under the given conditions.

A fair 6-sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number? $ \textbf{(A)}\dfrac16\qquad\textbf{(B)}\dfrac5{12}\qquad\textbf{(C)}\dfrac12\qquad\textbf{(D)}\dfrac7{12}\qquad\textbf{(E)}\dfrac56 $

If two dice are tossed, the probability that the product of the numbers showing on the tops of the dice is greater than $10$ is $\text{(A)}\ \dfrac{3}{7} \qquad \text{(B)}\ \dfrac{17}{36} \qquad \text{(C)}\ \dfrac{1}{2} \qquad \text{(D)}\ \dfrac{5}{8} \qquad \text{(E)}\ \dfrac{11}{12}$

When a fair six-sided dice is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the $5$ faces than can be seen is divisible by $6$? $\textbf{(A)}\ 1/3 \qquad \textbf{(B)}\ 1/2 \qquad \textbf{(C)}\ 2/3 \qquad \textbf{(D)}\ 5/6 \qquad \textbf{(E)}\ 1$

A cube is formed from $n^3$ ($n \ge 2$) unit cubes, each painted white on five randomly selected sides. This cube is dipped into paint remover and broken into the original unit cubes. What is the expected number of these unit cubes with exactly four sides painted white?

Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other? $ \textbf{(A)} \frac14 \qquad\textbf{(B)} \frac13 \qquad\textbf{(C)} \frac12 \qquad\textbf{(D)} \frac23 \qquad\textbf{(E)} \frac34 $

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

If $a$ and $b$ are selected uniformly from $\{0,1,\ldots,511\}$ without replacement, the expected number of $1$'s in the binary representation of $a+b$ can be written in simplest from as $\tfrac{m}{n}$. Compute $m+n$.

Let $k$ be an integer in the interval $[1,99]$. A fair coin is to be flipped $100$ times. Let $$\varepsilon_j =\begin{cases} 1, \text{if the j-th flip is a head} \\ 2, \text{f the j-th flip is a tail}\end{cases}$$ Let $M_k$ denote the probability that there exists a number $i$ such that $k+\varepsilon_1 +...+\varepsilon_i = 100$. How to choose $k$ so as to maximize the probability $M_k$?

A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\frac{a}{b}$, where $a$ and $b$ are relatively prime, compute $\lceil\log_{2}(100a+b)\rceil$. [i]Proposed by Lewis Chen[/i]

Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value. [i]Proposed by Marcin Kuczma, Poland[/i]

Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$? $\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) }16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20$

We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?

A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k=1,2,3,\ldots.$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? $\textbf{(A) } \frac{1}{4} \qquad\textbf{(B) } \frac{2}{7} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{3}{8} \qquad\textbf{(E) } \frac{3}{7}$

Tyler rolls two $ 4025 $ sided fair dice with sides numbered $ 1, \dots , 4025 $. Given that the number on the first die is greater than or equal to the number on the second die, what is the probability that the number on the first die is less than or equal to $ 2012 $?

A father, a mother and son hold a family tournament, playing a two person board game with no ties. The tournament rules are: (i) The weakest player chooses the first two contestants. (ii) The winner of any game plays the next game against the person left out. (iii) The first person to win two games wins the tournament. The father is the weakest player, the son the strongest, and it is assumed that any player's probability of winning an individual game from another player does not change during the tournament. Prove that the father's optimal strategy for winning the tournament is to play the first game with his wife.

The probability of drawing a red marble from a bag is $\frac{3}{5}$. After some red marbles are removed, the probability of drawing a red marble is $\frac{2}{7}$. What is the smallest number of marbles that could have originally been in the bag?

Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube? $ \textbf{(A)}\ \frac{1}{4} \qquad \textbf{(B)}\ \frac{3}{8} \qquad \textbf{(C)}\ \frac{4}{7} \qquad \textbf{(D)}\ \frac{5}{7} \qquad \textbf{(E)}\ \frac{3}{4}$

Liam writes the number $0$ on a board, then performs a series of turns. Each turn, he chooses a nonzero integer so that for every nonzero integer $N,$ he chooses $N$ with $3^{- |N|}$ probability. He adds his chosen integer $N$ to the last number written on the board, yielding a new number. He writes the new number on the board and uses it for the next turn. Liam repeats the process until either $8$ or $9$ is written on the board, at which point he stops. Given that Liam eventually stopped, find the probability the last number he wrote on the board was $9.$

The train schedule in Hummut is hopelessly unreliable. Train $A$ will enter Intersection $X$ from the west at a random time between $9:00$ am and $2:30$ pm; each moment in that interval is equally likely. Train $B$ will enter the same intersection from the north at a random time between $9:30$ am and $12:30$ pm, independent of Train $A$; again, each moment in the interval is equally likely. If each train takes $45$ minutes to clear the intersection, what is the probability of a collision today?

Given $n$ chairs around a circle which are marked with numbers from 1 to $n$ .There are $k$, $k \leq 4 \cdot n$ students sitting on those chairs .Two students are called neighbours if there is no student sitting between them. Between two neighbours students ,there are at less 3 chairs. Find the number of choices of $k$ chairs so that $k$ students can sit on those and the condition is satisfied.

A student will take an exam in which they have to solve three chosen problems by chance of a list of $10$ possible problems. It will be approved if it correctly resolves two problems. Considering that the student can solve five of the problems on the list and not know how to solve others, how likely is he to pass the exam?

Sally randomly chooses three different numbers from the set $\left\{1,2,\ldots,14\right\}$. What is the probability that the sum of her smallest number and her biggest number is at least $15$?

A certain college student had the night of February 23 to work on a chemistry problem set and a math problem set (both due on February 24, 2006). If the student worked on his problem sets in the math library, the probability of him finishing his math problem set that night is 95% and the probability of him finishing his chemistry problem set that night is 75%. If the student worked on his problem sets in the the chemistry library, the probability of him finishing his chemistry problem set that night is 90% and the probability of him finishing his math problem set that night is 80%. Since he had no bicycle, he could only work in one of the libraries on February 23rd. He works in the math library with a probability of 60%. Given that he finished both problem sets that night, what is the probability that he worked on the problem sets in the math library?