Ted flips seven fair coins. there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that Ted flips at least two heads given that he flips at least three tails. Find $m+n$.

Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \le a<b<c<d \le 10$, and $a+d>b+c$. How many ordered quadruples are there?

Let $ n$ be a positive integer congruent to $1$ modulo $4$. Xantippa has a bag of $n + 1$ balls numbered from $ 0$ to $n$. She draws a ball (randomly, equally distributed) from the bag and reads its number: $k$, say. She keeps the ball and then picks up another $k$ balls from the bag (randomly, equally distributed, without repossession). Finally, she adds up the numbers of all the $k + 1$ balls she picked up. What is the probability that the sum will be odd?

On a plane, there are $7$ seats. Each is assigned to a passenger. The passengers walk on the plane one at a time. The first passenger sits in the wrong seat (someone else's). For all the following people, they either sit in their assigned seat, or if it is full, randomly pick another. You are the last person to board the plane. What is the probability that you sit in your own seat?

A real number $ to $ is randomly and uniformly chosen from the $ [- 3,4] $ interval. What is the probability that all roots of the polynomial $ x ^ 3 + ax ^ 2 + ax + 1 $ are real?

We are given $b$ white balls and $n$ black balls ($b,n>0$) which are to be distributed among two urns, at least one in each. Let $s$ be the number of balls in the first urn, and $r$ the number of white ones among them. One randomly chooses an urn and randomly picks a ball from it. (a) Compute the probability $p$ that the drawn ball is white. (b) If $s$ is fixed, for which $r$ is $p$ maximal? (c) Find the distribution of balls among the urns which maximizes $p$. (d) Give a generalization for larger numbers of colors and urns.

Let $\zeta_1, \ldots, \zeta_n$ be (not necessarily independent) random variables with normal distribution for which $E\zeta_j=0$ and $E\zeta_j^2\le 1$ for all $1\le j\le n$. Prove that $$E\left( \max_{1\le j\le n} \zeta_j \right)\le\sqrt{2\log n}$$ (translated by Miklós Maróti)

An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color? $\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12$

Suppose there are 8 white balls and 2 red balls in a packet. Each time one ball is drawn and replaced by a white one. Find the probability that the last red ball is drawn in the fourth draw.

A coin that comes up heads with probability $ p > 0$ and tails with probability $ 1\minus{}p > 0$ independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to $ \frac{1}{25}$ of the probability of five heads and three tails. Let $ p \equal{} \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}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 $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.

If I roll three fair $4$-sided dice, what is the probability that the sum of the resulting numbers is relatively prime to the product of the resulting numbers?

One of the boxes that Joshua and Wendy unpack has Joshua's collection of board games. Michael, Wendy, Alexis, and Joshua decide to play one of them, a game called $\textit{Risk}$ that involves rolling ordinary six-sided dice to determine the outcomes of strategic battles. Wendy has never played before, so early on Michael explains a bit of strategy. "You have the first move and you occupy three of the four territories in the Australian continent. You'll want to attack Joshua in Indonesia so that you can claim the Australian continent which will give you bonus armies on your next turn." "Don't tell her $\textit{that!}$" complains Joshua. Wendy and Joshua begin rolling dice to determine the outcome of their struggle over Indonesia. Joshua rolls extremely well, overcoming longshot odds to hold off Wendy's attack. Finally, Wendy is left with one chance. Wendy and Joshua each roll just one six-sided die. Wendy wins if her roll is $\textit{higher}$ than Joshua's roll. Let $a$ and $b$ be relatively prime positive integers so that $a/b$ is the probability that Wendy rolls higher, giving her control over the continent of Australia. Find the value of $a+b$.

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]

There are $100$ students who want to sign up for the class Introduction to Acting. There are three class sections for Introduction to Acting, each of which will fit exactly $20$ students. The $100$ students, including Alex and Zhu, are put in a lottery, and 60 of them are randomly selected to fill up the classes. What is the probability that Alex and Zhu end up getting into the same section for the class?

Katie begins juggling five balls. After every second elapses, there is a chance she will drop a ball. If she is currently juggling $ k$ balls, this probability is $ \frac{k}{10}$. Find the expected number of seconds until she has dropped all the balls.

Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle? $ \textbf{(A)}\ \frac{1}{36} \qquad \textbf{(B)}\ \frac{1}{24} \qquad \textbf{(C)}\ \frac{1}{18} \qquad \textbf{(D)}\ \frac{1}{12} \qquad \textbf{(E)}\ \frac{1}{9}$

There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$, and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.

Leo and Paul are at the Berkeley BART station and are racing to San Francisco. Leo is planning to take the line that takes him directly to SF, and because he has terrible BART luck, his train will arrive in some integer number of minutes, with probability $\frac i{210}$ for $1\le i\le20$ at any given minute. Paul will take a second line, whose trains always arrive before Leo’s train, with uniform probability. However, Paul must also make a transfer to a 3rd line, whose trains arrive with uniform probability between $0$ and $10$ minutes after Paul reaches the transfer station. What is the probability that Leo gets to SF before Paul does?

A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white? $\textbf{(A) }\frac{5}{54}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{5}{27}\qquad\textbf{(D) }\frac{2}{9}\qquad \textbf{(E) }\frac{1}{3}$

Two persons, $X$ and $Y$ , play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. For each player determine the probability of winning a match within $5$ games. Determine the probabilities of winning in an unlimited number of games. If $X$ bets $1$, how much must $Y$ bet for the game to be fair ?

Dan has a fair $6$-sided die with faces labeled $1,2,3,4,+,$ and $-.$ In order to complete the equation \[ \underline{\qquad} \ \underline{\qquad} \ \underline{\qquad}=\underline{\qquad}, \] Dan repeatedly rolls his die and fills in a blank with the character he obtained, starting with the leftmost blank and progressing rightward. The probability that, when all blanks are filled, Dan forms a true equation, is $\frac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p+q.$

In fencing, you win a round if you are the first to reach $15$ points. Suppose that when $A$ plays against $B$, at any point during the round, $A$ scores the next point with probability $p$ and $B$ scores the next point with probability $q=1-p$. (However, they never can both score a point at the same time.) Suppose that in this round, $A$ already has $14-k$ points, and $B$ has $14-\ell$ (where $0\le k,\ell\le 14$). By how much will the probability that $A$ wins the round increase if $A$ scores the next point?

A set of lines in the plane is in [i]general position[/i] if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its [i]finite regions[/i]. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ lines blue in such a way that none of its finite regions has a completely blue boundary. [i]Note[/i]: Results with $\sqrt{n}$ replaced by $c\sqrt{n}$ will be awarded points depending on the value of the constant $c$.