My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what is the expected score of the losing team? $\textbf{(A) }2/3\hspace{14em}\textbf{(B) }1\hspace{14.8em}\textbf{(C) }3/2$ $\textbf{(D) }8/5\hspace{14em}\textbf{(E) }5/8\hspace{14em}\textbf{(F) }2$ $\textbf{(G) }0\hspace{14.9em}\textbf{(H) }5/2\hspace{14em}\textbf{(I) }2/5$ $\textbf{(J) }3/4\hspace{14em}\,\textbf{(K) }4/3\hspace{13.9em}\textbf{(L) }2007$

$n$ pairs are formed from $n$ girls and $n$ boys at random. What is the probability of having at least one pair of girls? For which $n$ the probability is over $0,9?$

Two dice appear to be standard dice with their faces numbered from $1$ to $6$, but each die is weighted so that the probability of rolling the number $k$ is directly proportional to $k$. The probability of rolling a $7$ with this pair of dice is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

$n, k$ are positive integers. $A_0$ is the set $\{1, 2, ... , n\}$. $A_i$ is a randomly chosen subset of $A_{i-1}$ (with each subset having equal probability). Show that the expected number of elements of $A_k$ is $\dfrac{n}{2^k}$

Rectangle $ABCD$ has side lengths $AB = 6\sqrt3$ and $BC = 8\sqrt3$. The probability that a randomly chosen point inside the rectangle is closer to the diagonal $\overline{AC}$ than to the outside of the rectangle is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $n$ be a integer and $n \ge 3$, and $T_1T_2...T_n$ is a regular n-gon. Distinct $3$ points $T_i , T_j , T_k$ are chosen randomly. Determine the probability of triangle $T_iT_jT_k$ being an acute triangle.

Let $ \mu$ and $ \nu$ be two probability measures on the Borel sets of the plane. Prove that there are random variables $ \xi_1, \xi_2, \eta_1, \eta_2$ such that (a) the distribution of $ (\xi_1, \xi_2)$ is $ \mu$ and the distribution of $ (\eta_1, \eta_2)$ is $ \nu$, (b) $ \xi_1 \leq \eta_1, \xi_2 \leq \eta_2$ almost everywhere, if an only if $ \mu(G) \geq \nu(G)$ for all sets of the form $ G\equal{}\cup_{i\equal{}1}^k (\minus{}\infty, x_i) \times (\minus{}\infty, y_i).$ [i]P. Major[/i]

An angle $x$ is chosen at random from the interval $0^\circ < x < 90^\circ$. Let $p$ be the probability that the numbers $\sin^2 x$, $\cos^2 x$, and $\sin x \cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n$, where $d$ is the number of degrees in $\arctan m$ and $m$ and $n$ are positive integers with $m + n < 1000$, find $m + n$.

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

Let $ P$ be a probability distribution defined on the Borel sets of the real line. Suppose that $ P$ is symmetric with respect to the origin, absolutely continuous with respect to the Lebesgue measure, and its density function $ p$ is zero outside the interval $ [\minus{}1,1]$ and inside this interval it is between the positive numbers $ c$ and $ d$ ($ c < d$). Prove that there is no distribution whose convolution square equals $ P$. [i]T. F. Mori, G. J. Szekely[/i]

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?

Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select $4$ points on the segment $P_0P_5$, and immediately after selecting those points, we label those $4$ selected points $P_1, P_2, P_3, P_4$ from left to right. Similarly, we randomly select $4$ points on the segment $Q_0Q_5$, and immediately after selecting those points, we label those $4$ points $Q_1, Q_2, Q_3, Q_4$ from left to right. The segments $P_1Q_1, P_2Q_2, P_3Q_3, P_4Q_4$ divide the rectangular chocolate bar into $5$ smaller trapezoidal pieces of chocolate. The probability that exactly $3$ pieces of chocolate contain both dark and white chocolate can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [Diagram in the individuals file for this exam on the Chmmc website]

There are 2014 houses in a circle. Let $A$ be one of these houses. Santa Claus enters house $A$ and leaves a gift. Then with probability $1/2$ he visits $A$'s left neighbor and with probability $1/2$ he visits $A$'s right neighbor. He leaves a gift also in that second house, and then repeats the procedure (visits with probability $1/2$ either of the neighbors, leaves a gift, etc). Santa finishes as soon as every house has received at least one gift. Prove that any house $B$ different from $A$ has a probability of $1/2013$ of being the last house receiving a gift.

For a randomly selected permutation $ \mathbf{f} = (f_1,..., f_n) $ of the set $ \{1,\ldots, n\} $ let us denote by $ X(\mathbf{f}) $ the largest number $ k \leq n $ such that $ f_i < f_{ i+1} $ for all numbers $ i < k $. Prove that the expected value of the random variable $ X $ is $ \sum_{k=1}^n \frac{1}{k!} $.

How many ways are there to cover a $3\times 8$ rectangle with $12$ identical dominoes?

For which integers between $2000$ and $2010$ (including) is the probability that a random divisor is smaller or equal $45$ the largest?

DeAndre Jordan shoots free throws that are worth $1$ point each. He makes $40\%$ of his shots. If he takes two shots find the probability that he scores at least $1$ point.

Call a positive rational number in simplest terms [i]coddly[/i] if its numerator and denominator are both odd. Consider the equation $$2017= x_1\text{ }\square\text{ }x_2\text{ }\square\text{ }x_3\text{ }\ldots \text{ }\square \text{ }x_{2016} \text{ }\square \text{ }x_{2017},$$ where there are $2016$ boxes. We fill up the boxes randomly with the operations $+$, $-$, and $\times$. Compute the probability that there exists a solution in [b]distinct[/b] coddly numbers $(x_1,x_2, \ldots x_{2017})$ to the resulting equation.

There is a bag with balls whose colors are $c_1, c_2, \dots, c_n$. Let $a_i$ be the number of balls inside the bag with color $c_i$. We are drawing $n$ balls from the bag one by one with replacement. If $p(a_1,a_2,\dots, a_n)$ denotes the probability that at least two of them have same color, which one below is smaller? $\textbf{(A)}\ p(2,2,2,1) \qquad\textbf{(B)}\ p(1,1,1,1) \qquad\textbf{(C)}\ p(2,2,3) \qquad\textbf{(D)}\ p(2,2,1) \qquad\textbf{(E)}\ p(1,1,1)$

$6$ people each have a hat. If they shuffle their hats and redistribute them, what is the probability that exactly one person gets their own hat back?

A traffic light runs repeatedly through the following cycle: green for $ 30$ seconds, then yellow for $ 3$ seconds, and then red for $ 30$ seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? $ \textbf{(A)}\ \frac {1}{63}\qquad \textbf{(B)}\ \frac {1}{21}\qquad \textbf{(C)}\ \frac {1}{10}\qquad \textbf{(D)}\ \frac {1}{7}\qquad \textbf{(E)}\ \frac {1}{3}$

Two dice are thrown. What is the probability that the product of the two numbers is a multiple of 5? $ \text{(A)}\ \frac{1}{36}\qquad\text{(B)}\ \frac{1}{18}\qquad\text{(C)}\ \frac{1}{6}\qquad\text{(D)}\ \frac{11}{36}\qquad\text{(E)}\ \frac{1}{3} $

In a skiing jump competition $65$ contestants take part. They jump with the previously established order. Each of them jumps once. We assume that the obtained results are different and the orders of the contestants after the competition are equally likely. In each moment of the competition by a leader we call a person who is scored the best at this moment. Denote by $p$ the probability that during the whole competition there was exactly one change of the leader. Prove that $p > 1/16$.

Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6$. What is the probability that Diana's number is larger than Apollo's number? $\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{5}{12} \qquad \text{(C)}\ \dfrac{4}{9} \qquad \text{(D)}\ \dfrac{17}{36} \qquad \text{(E)}\ \dfrac{1}{2}$

Suppose we choose two numbers $x,y\in[0,1]$ uniformly at random. If the probability that the circle with center $(x,y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1]\times [0,1]$ is written as $\tfrac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?