Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral.

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 $a+b$.

$n, k$ 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 $\dfrac{n}{2^k}$

Among $100$ points in the plane, no three collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points whose labels differ by at least $50$. [i]Proposed by Evan Chen[/i]

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.

What is the probability that a randomly drawn positive factor of $ 60$ is less than $ 7$? $ \textbf{(A)}\ \frac{1}{10} \qquad \textbf{(B)}\ \frac{1}{6} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{1}{3} \qquad \textbf{(E)}\ \frac{1}{2}$

Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as $\frac{1}{2^n}$ for some positive integer $n$. Find $n$.

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 $n,k$ be given positive integers with $n>k$. Prove that: \[ \frac{1}{n+1} \cdot \frac{n^n}{k^k (n-k)^{n-k}} < \frac{n!}{k! (n-k)!} < \frac{n^n}{k^k(n-k)^{n-k}} \]

Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine? $ \textbf{(A)}\ \frac{7}{11}\qquad\textbf{(B)}\ \frac{9}{13}\qquad\textbf{(C)}\ \frac{11}{15}\qquad\textbf{(D)}\ \frac{15}{19}\qquad\textbf{(E)}\ \frac{15}{16} $

Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3? $\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{49}{216}\qquad\textbf{(C) }\frac{25}{108}\qquad\textbf{(D) }\frac{17}{72}\qquad\textbf{(E) }\frac{13}{54}$

A palindrome between $ 1000$ and $ 10,000$ is chosen at random. What is the probability that it is divisible by $ 7?$ $ \textbf{(A)}\ \dfrac{1}{10} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{7} \qquad \textbf{(D)}\ \dfrac{1}{6}\qquad \textbf{(E)}\ \dfrac{1}{5}$

An $m\times n$ checkerboard is colored randomly: each square is independently assigned red or black with probability $\frac12.$ we say that two squares, $p$ and $q$, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starting at $p$ and ending at $q,$ in which successive squares in the sequence share a common side. Show that the expected number of connected monochromatic regions is greater than $\frac{mn}8.$

Eight points are spaced around at intervals of one unit around a $2 \times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart? [asy] size((50)); dot((5,0)); dot((5,5)); dot((0,5)); dot((-5,5)); dot((-5,0)); dot((-5,-5)); dot((0,-5)); dot((5,-5)); [/asy] $ \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{2}{7}\qquad\textbf{(C)}\ \frac{4}{11}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{4}{7} $

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)

There are two cards; one is red on both sides and the other is red on one side and blue on the other. The cards have the same probability (1/2) of being chosen, and one is chosen and placed on the table. If the upper side of the card on the table is red, then the probability that the under-side is also red is $ \textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac12 \qquad \textbf{(D)}\ \frac23 \qquad \textbf{(E)}\ \frac34$

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]

Hour part of a defective digital watch displays only the numbers from $1$ to $12$. After one minute from $ n: 59$, although it must display $ \left(n \plus{} 1\right): 00$, it displays $ 2n: 00$ (Think in $ mod\, 12$). For example, after $ 7: 59$, it displays $ 2: 00$ instead of $ 8: 00$. If we set the watch to an arbitrary time, what is the probability that hour part displays $4$ after exactly one day? $\textbf{(A)}\ \frac {1}{12} \qquad\textbf{(B)}\ \frac {1}{4} \qquad\textbf{(C)}\ \frac {1}{3} \qquad\textbf{(D)}\ \frac {1}{2} \qquad\textbf{(E)}\ \text{None}$

In a tennis ball factory there are $4$ machines $m_1 , m_2 , m_3 , m_4$, which produce, respectively, $10\%$, $20\%$, $30\%$ and $40\%$ of the balls that come out of the factory. The machine $m_1$ introduces defects in $1\%$ of the balls it manufactures, the machine $m_2$ in $2\%$, $m_3$ in $4\%$ and $m_4$ in $15\%$. Of the balls manufactured In one day, one is chosen at random and it turns out to be defective. What is the probability that Has this ball been made by the machine $ m_3$ ?

Integers $ a,$ $ b,$ $ c,$ and $ d,$ not necessarily distinct, are chosen independantly and at random from $ 0$ to $ 2007,$ inclusive. What is the probability that $ ad \minus{} bc$ is even? $ \textbf{(A)}\ \frac {3}{8}\qquad \textbf{(B)}\ \frac {7}{16}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {9}{16}\qquad \textbf{(E)}\ \frac {5}{8}$

From a given triangle of unit area, we choose two points independetly with uniform distribution. The straight line connecting these points divides the triangle. with probability one, into a triangle and a quadrilateral. Calculate the expected values of the areas of these two regions. [A. Renyi]

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

For each positive integer $ n$, evaluate the sum \[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]

A coin is altered so that the probability that it lands on heads is less than $ \frac {1}{2}$ and when the coin is flipped four times, the probability of an equal number of heads and tails is $ \frac {1}{6}$. What is the probability that the coin lands on heads? $ \textbf{(A)}\ \frac {\sqrt {15} \minus{} 3}{6}\qquad \textbf{(B)}\ \frac {6 \minus{} \sqrt {6\sqrt {6} \plus{} 2}}{12}\qquad \textbf{(C)}\ \frac {\sqrt {2} \minus{} 1}{2}\qquad \textbf{(D)}\ \frac {3 \minus{} \sqrt {3}}{6}\qquad \textbf{(E)}\ \frac {\sqrt {3} \minus{} 1}{2}$

Three points are chosen randomly and independently on a circle. The probability that all three pairwise distance between the points are less than the radius of the circle is $\frac{1}{K}$, $K\in\mathbb{N}$. Find $K$.