Two teams are in a best-two-out-of-three playoff: the teams will play at most $3$ games, and the winner of the playoff is the first team to win $2$ games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a $\frac{2}{3}$ chance of winning at home, and its probability of winning when playing away from home is $p$. Outcomes of the games are independent. The probability that Team A wins the playoff is $\frac{1}{2}$. Then $p$ can be written in the form $\frac{1}{2}(m - \sqrt{n})$, where $m$ and $n$ are positive integers. What is $m + n$? $\textbf{(A) } 10 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14$

A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least $x$ times as large as the shorter piece is $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{2}{x}\qquad\textbf{(C) }\frac{1}{x+1}\qquad\textbf{(D) }\frac{1}{x}\qquad \textbf{(E) }\frac{2}{x+1}$

Let $\zeta_1, \zeta_2,\ldots$ be identically distributed, independent real-valued random variables with expected value $0$. Suppose that the $\Lambda (\lambda) := \log \mathbb E \exp (\lambda \zeta_i)$ logarithmic moment-generating function always exists for $\lambda\in\mathbb R$ ($\mathbb E$ is the expected value). Furthermore, let $G\colon\mathbb R \rightarrow \mathbb R$ be a function such that $G(x)\leq \min (|x|, x^2)$. Prove that for small $\gamma >0$ the following sequence is bounded: $$\left\{ \mathbb E \exp \left( \gamma l G \left( \frac 1l (\zeta_1+\ldots + \zeta_l)\right)\right)\right\}^{\infty}_{l=1}$$ (translated by j___d)

Let $x_0=1$, and let $\delta$ be some constant satisfying $0<\delta<1$. Iteratively, for $n=0,1,2,\dots$, a point $x_{n+1}$ is chosen uniformly form the interval $[0,x_n]$. Let $Z$ be the smallest value of $n$ for which $x_n<\delta$. Find the expected value of $Z$, as a function of $\delta$.

Daredevil Darren challenges Forgetful Fred to spell "Johns Hopkins." Forgetful Fred will spell it correctly except for the 's's; there is a $\frac{1}{3}$ and $\frac{1}{4}$ chance that he will omit the 's' in the first and last names, respectively, with his mistakes being independent of each other. If Forgetful Fred spells the name correctly, then he is happy; otherwise, Daredevil Darren will present him with a dare, and there is a $\frac{9}{10}$ chance that Forgetful Fred will not be happy. Find the probability that Forgetful Fred will be happy.

Three students, with different names, line up single file. What is the probability that they are in alphabetical order from front-to-back? $\textbf{(A)}\ \dfrac{1}{12} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{6} \qquad \textbf{(D)}\ \dfrac{1}{3} \qquad \textbf{(E)}\ \dfrac{2}{3}$

A fair coin is tossed repeatedly until there is a run of an odd number of heads followed by a tail. Determine the expected number of tosses.

Two different numbers are randomly selected from the set ${ - 2, -1, 0, 3, 4, 5}$ and multiplied together. What is the probability that the product is $0$? $\textbf{(A) }\dfrac{1}{6}\qquad\textbf{(B) }\dfrac{1}{5}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad \textbf{(E) }\dfrac{1}{2}$

Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7$. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice? $\textbf{(A) } \frac{7}{36} \qquad\textbf{(B) } \frac{5}{24} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{17}{72} \qquad\textbf{(E) } \frac{1}{4}$

There are $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find $|P_s-P_d|$. $\textbf{(A) }0\qquad\textbf{(B) }\dfrac1{2002}\qquad\textbf{(C) }\dfrac1{2001}\qquad\textbf{(D) }\dfrac2{2001}\qquad\textbf{(E) }\dfrac1{1000}$

The George Washington Bridge is $2016$ meters long. Sally is standing on the George Washington Bridge, $1010$ meters from its left end. Each step, she either moves $1$ meter to the left or $1$ meter to the right, each with probability $\dfrac{1}{2}$. What is the expected number of steps she will take to reach an end of the bridge?

How many non-empty subsets $ S$ of $ \{1, 2, 3, \ldots, 15\}$ have the following two properties? (1) No two consecutive integers belong to $ S$. (2) If $ S$ contains $ k$ elements, then $ S$ contains no number less than $ k$. $ \textbf{(A) } 277\qquad \textbf{(B) } 311\qquad \textbf{(C) } 376\qquad \textbf{(D) } 377\qquad \textbf{(E) } 405$

Spinners A and B are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even? [asy] defaultpen(linewidth(1)); draw(unitcircle); draw((1,0)--(-1,0)); draw((0,1)--(0,-1)); draw(shift(3,0)*unitcircle); draw(shift(3,0)*(origin--dir(90))); draw(shift(3,0)*(origin--dir(210))); draw(shift(3,0)*(origin--dir(330))); draw(0.7*dir(200)--0.7*dir(20), linewidth(0.7), EndArrow(7)); draw(shift(3,0)*(0.7*dir(180+65)--0.7*dir(65)), linewidth(0.7), EndArrow(7)); label("$1$", (-0.45,0.1), N); label("$4$", (-0.45,-0.1), S); label("$3$", (0.45,-0.1), S); label("$2$", (0.45,0.1), N); label("$1$", shift(3,0)*(-0.25,0.1), NW); label("$2$", shift(3,0)*(0.25,0.1), NE); label("$3$", shift(3,0)*(0,-0.3), S); label("$A$", (0,-1), S); label("$B$", (3,-1), S); [/asy] $ \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{3}{4} $

Let $K$ be a finite field with $q$ elements and let $n \ge q$ be an integer. Find the probability that by choosing an $n$-th degree polynomial with coefficients in $K,$ it doesn't have any root in $K.$

If two points are selected at random on a fixed circle and the chord between the two points is drawn, what is the probability that its length exceeds the radius of the circle?

Bob writes a random string of $5$ letters, where each letter is either $A, B, C,$ or $D$. The letter in each position is independently chosen, and each of the letters $A, B, C, D$ is chosen with equal probability. Given that there are at least two $A's$ in the string, find the probability that there are at least three $A's$ in the string.

Starting at $(0, 0)$, Richard takes $2n+1$ steps, with each step being one unit either East, North, West, or South. For each step, the direction is chosen uniformly at random from the four possibilities. Determine the probability that Richard ends at $(1, 0)$.

For each real number $a$ with $0 \leq a \leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$, respectively, and let $P(a)$ be the probability that $$\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1.$$ What is the maximum value of $P(a)?$ $\textbf{(A)}\ \frac{7}{12} \qquad\textbf{(B)}\ 2 - \sqrt{2} \qquad\textbf{(C)}\ \frac{1+\sqrt{2}}{4} \qquad\textbf{(D)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(E)}\ \frac{5}{8}$

A particle performing a random walk on the integer points of the semi-axis $x \ge 0$ moves a distance $1$ to the right with probability $a$, and to the left with probability $b$, and stands still in the remaining cases (if $x = 0$, it stands still instead of moving to the left). Determine the steady-state probability distribution, and also the expectation of $x$ and $x^2$ over a long time, if the particle starts at the point $0$.

Ten birds land on a $10$-meter-long wire, each at a random point chosen uniformly along the wire. (That is, if we pick out any $x$-meter portion of the wire, there is an $\tfrac{x}{10}$ probability that a given bird will land there.) What is the probability that every bird sits more than one meter away from its closest neighbor?

A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of \[ \sum_{i = 1}^{2012} | a_i - i |, \] then compute the sum of the prime factors of $S$. [i]Proposed by Aaron Lin[/i]

Integers $a$, $b$, $c$ are selected independently and at random from the set $ \{ 1, 2, \cdots, 10 \} $, with replacement. If $p$ is the probability that $a^{b-1}b^{c-1}c^{a-1}$ is a power of two, compute $1000p$. [i]Proposed by Evan Chen[/i]

An urn contain a number of yellow and green balls. You extract two balls from the urn (without adding them back) and calculate the probability of both balls being green. Can you choose the number of yellow and green balls such that this probability to be $\frac{1}{4}$?

Maya lists all the positive divisors of $ 2010^2$. She then randomly selects two distinct divisors from this list. Let $ p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $ p$ can be expressed in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

Byan rolls a $12$-sided die, a $14$-sided die, a $20$-sided die, and a $24$-sided die. The probability the sum of the numbers the die landed on is divisible by $7$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Lightning 3.1[/i]