Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two chords intersect? $ \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2} $

Rachel has the number $1000$ in her hands. When she puts the number $x$ in her left pocket, the number changes to $x+1.$ When she puts the number $x$ in her right pocket, the number changes to $x^{-1}.$ Each minute, she flips a fair coin. If it lands heads, she puts the number into her left pocket, and if it lands tails, she puts it into her right pocket. She then takes the new number out of her pocket. If the expected value of the number in Rachel's hands after eight minutes is $E,$ compute $\left\lfloor\frac{E}{10}\right\rfloor.$

Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1,2, \dots,20$ on its sides). He conceals the results but tells you that at least half the rolls are $20$. Suspicious, you examine the first two dice and find that they show $20$ and $19$ in that order. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show $20$?

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.

$32$ teams, ranked $1$ through $32$, enter a basketball tournament that works as follows: the teams are randomly paired and in each pair, the team that loses is out of the competition. The remaining $16$ teams are randomly paired, and so on, until there is a winner. A higher ranked team always wins against a lower-ranked team. If the probability that the team ranked $3$ (the third-best team) is one of the last four teams remaining can be written in simplest form as $\dfrac{m}{n}$, compute $m+n$.

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

Let $a/b$ be the probability that a randomly selected divisor of $2007$ is a multiple of $3$. If $a$ and $b$ are relatively prime positive integers, find $a+b$.

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? $\textbf{(A) } \frac{1}{4} \qquad\textbf{(B) } \frac{2}{7} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{3}{8} \qquad\textbf{(E) } \frac{3}{7}$

In triangle $ ABC$, $ AB \equal{} AC \equal{} 100$, and $ BC \equal{} 56$. Circle $ P$ has radius $ 16$ and is tangent to $ \overline{AC}$ and $ \overline{BC}$. Circle $ Q$ is externally tangent to $ P$ and is tangent to $ \overline{AB}$ and $ \overline{BC}$. No point of circle $ Q$ lies outside of $ \triangle ABC$. The radius of circle $ Q$ can be expressed in the form $ m \minus{} n\sqrt {k}$, where $ m$, $ n$, and $ k$ are positive integers and $ k$ is the product of distinct primes. Find $ m \plus{} nk$.

A dog has three trainers: [list] [*]The first trainer gives him a treat right away. [*]The second trainer makes him jump five times, then gives him a treat. [*]The third trainer makes him jump three times, then gives him no treat. [/list] The dog will keep picking trainers with equal probability until he gets a treat. (The dog's memory isn't so good, so he might pick the third trainer repeatedly!) What is the expected number of times the dog will jump before getting a treat?

A point $ (x,y)$ is randomly picked from inside the rectangle with vertices $ (0,0)$, $ (4,0)$, $ (4,1)$, and $ (0,1)$. What is the probability that $ x<y$? $ \textbf{(A)}\ \frac{1}{8} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{3}{8} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{3}{4}$

Wendy takes Honors Biology at school, a smallish class with only fourteen students (including Wendy) who sit around a circular table. Wendy's friends Lucy, Starling, and Erin are also in that class. Last Monday none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarn would intersect if possible. If all possible positions of the students around the table are equally likely, let $m/n$ be the probability that the yarns intersect, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

The figure on the right shows the map of Squareville, where each city block is of the same length. Two friends, Alexandra and Brianna, live at the corners marked by $A$ and $B$, respectively. They start walking toward each other's house, leaving at the same time, walking with the same speed, and independently choosing a path to the other's house with uniform distribution out of all possible minimum-distance paths [that is, all minimum-distance paths are equally likely]. What is the probability they will meet? [asy] size(200); defaultpen(linewidth(0.8)); for(int i=0;i<=2;++i) { for(int j=0;j<=4;++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle); } } for(int i=3;i<=4;++i) { for(int j=3;j<=6;++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle); } } label("$A$",origin,SW); label("$B$",(5,7),SE); [/asy]

A modern artist paints all of his paintings by dividing his $3$ ft by $5$ ft canvas into $21$ random regions. He then colours some of the regions, and leaves some of them white. If the smallest region has area $a = 10$ square inches, and the probability that any given region with area $a_i$ is left white is $\frac{a}{a_i}$, then what is the probability that any given point on the canvas is left white? ($1$ ft $= 12$ in)

Three persons $A,B,C$, are playing the following game: A $k$-element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$. For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)

Let $x,y$ and $z$ be real numbers such that $0 \leq x,y,z \leq 1.$ Prove that \[xyz+(1-x)(1-y)(1-z) \leq 1.\] When does equality hold?

A point $ P$ is chosen at random in the interior of equilateral triangle $ ABC$. What is the probability that $ \triangle ABP$ has a greater area than each of $ \triangle ACP$ and $ \triangle BCP$? $ \textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{1}{3} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{2}{3}$

The Cincinnati Reals are playing the Houston Alphas in the last game of the Swirled Series. The Alphas are leading by $1$ run in the bottom of the $9\text{th}$ (last) inning, and the Reals are at bat. Each batter has a $\dfrac{1}{3}$ chance of hitting a single and a $\dfrac{2}{3}$ chance of making an out. If the Reals hit $5$ or more singles before they make $3$ outs, they will win. If the Reals hit exactly $4$ singles before they make $3$ outs, they will tie the game and send it into extra innings, and they will have a $\dfrac{3}{5}$ chance of eventually winning the game (since they have the added momentum of coming from behind). If the Reals hit fewer than $4$ singles, they will LOSE! What is the probability that the Alphas hold off the Reals and win, sending the packed Alphadome into a frenzy? Express the answer as a fraction.

Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Justin is being served two different types of chips, A-chips, and B-chips. If there are $3$ B-chips and $5$ A-chips, and if Justin randomly grabs $3$ chips, what is the probability that none of them are A-chips?

Two standard six-sided dice are tossed. What is the probability that the sum of the numbers is greater than $7$? $\text{(A) }1\qquad\text{(B) }\frac{5}{12}\qquad\text{(C) }\frac{2}{3}\qquad\text{(D) }\frac{4}{9}\qquad\text{(E) }\frac{7}{36}$

A candy company makes $5$ colors of jellybeans, which come in equal proportions. If I grab a random sample of $5$ jellybeans, what is the probability that I get exactly $2$ distinct colors?

Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D

A trawler is about to fish in territorial waters of a neighboring country, for what he has no licence. Whenever he throws the net, the coast-guard may stop him with the probability $1/k$, where $k$ is a fixed positive integer. Each throw brings him a fish landing of a fixed weight. However, if the coast-guard stops him, they will confiscate his entire fish landing and demand him to leave the country. The trawler plans to throw the net $n$ times before he returns to territorial waters in his country. Find $n$ for which his expected profit is maximal.

Jacob and Aaron are playing a game in which Aaron is trying to guess the outcome of an unfair coin which shows heads $\tfrac{2}{3}$ of the time. Aaron randomly guesses ``heads'' $\tfrac{2}{3}$ of the time, and guesses ``tails'' the other $\tfrac{1}{3}$ of the time. If the probability that Aaron guesses correctly is $p$, compute $9000p$. [i]Proposed by Aaron Lin[/i]