Integers $ a$, $ b$, $ c$, and $ d$, not necessarily distinct, are chosen independently 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}$

A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line? [asy]unitsize(.5cm); defaultpen(linewidth(.8pt)); dotfactor=3; pair[] dotted={(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)}; dot(dotted);[/asy]$ \textbf{(A)}\ \frac {1}{21}\qquad \textbf{(B)}\ \frac {1}{14}\qquad \textbf{(C)}\ \frac {2}{21}\qquad \textbf{(D)}\ \frac {1}{7}\qquad \textbf{(E)}\ \frac {2}{7}$

James writes down fifteen 1's in a row and randomly writes $+$ or $-$ between each pair of consecutive 1's. One such example is \[1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.\] What is the probability that the value of the expression James wrote down is $7$? $\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l} \textbf{(A) }0&\textbf{(B) }\dfrac{6435}{2^{14}}&\textbf{(C) }\dfrac{6435}{2^{13}}\\\\ \textbf{(D) }\dfrac{429}{2^{12}}&\textbf{(E) }\dfrac{429}{2^{11}}&\textbf{(F) }\dfrac{429}{2^{10}}\\\\ \textbf{(G) }\dfrac1{15}&\textbf{(H) } \dfrac1{31}&\textbf{(I) }\dfrac1{30}\\\\ \textbf{(J) }\dfrac1{29}&\textbf{(K) }\dfrac{1001}{2^{15}}&\textbf{(L) }\dfrac{1001}{2^{14}}\\\\ \textbf{(M) }\dfrac{1001}{2^{13}}&\textbf{(N) }\dfrac1{2^7}&\textbf{(O) }\dfrac1{2^{14}}\\\\ \textbf{(P) }\dfrac1{2^{15}}&\textbf{(Q) }\dfrac{2007}{2^{14}}&\textbf{(R) }\dfrac{2007}{2^{15}}\\\\ \textbf{(S) }\dfrac{2007}{2^{2007}}&\textbf{(T) }\dfrac1{2007}&\textbf{(U) }\dfrac{-2007}{2^{14}}\end{array}$

A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0),$ $(2020, 0),$ $(2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth$?$ $\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7$

Let $x$ be a real number selected uniformly at random between 100 and 200. If $\lfloor {\sqrt{x}} \rfloor = 12$, find the probability that $\lfloor {\sqrt{100x}} \rfloor = 120$. ($\lfloor {v} \rfloor$ means the greatest integer less than or equal to $v$.) $\text{(A)} \ \frac{2}{25} \qquad \text{(B)} \ \frac{241}{2500} \qquad \text{(C)} \ \frac{1}{10} \qquad \text{(D)} \ \frac{96}{625} \qquad \text{(E)} \ 1$

A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out. What is the probability that all beads in the bag are red after three such replacements? $ \textbf{(A)}\ \frac{1}{8} \qquad \textbf{(B)}\ \frac{5}{32} \qquad \textbf{(C)}\ \frac{9}{32} \qquad \textbf{(D)}\ \frac{3}{8} \qquad \textbf{(E)}\ \frac{7}{16}$

From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon? [asy] size(3cm); pair A[]; for (int i=0; i<9; ++i) { A[i] = rotate(22.5+45*i)*(1,0); } filldraw(A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--A[6]--A[7]--cycle,gray,black); for (int i=0; i<8; ++i) { dot(A[i]); } [/asy] $\textbf{(A) } \frac{2}{7} \qquad \textbf{(B) } \frac{5}{42} \qquad \textbf{(C) } \frac{11}{14} \qquad \textbf{(D) } \frac{5}{7} \qquad \textbf{(E) } \frac{6}{7}$

Let $n$ be a positive integer and let vectors $v_1$, $v_2$, $\ldots$, $v_n$ be given in the plain. A flea originally sitting in the origin moves according to the following rule: in the $i$th minute (for $i=1,2,\ldots,n$) it will stay where it is with probability $1/2$, moves with vector $v_i$ with probability $1/4$, and moves with vector $-v_i$ with probability $1/4$. Prove that after the $n$th minute there exists no point which is occupied by the flea with greater probability than the origin. [i]Proposed by Péter Pál Pach, Budapest[/i]

In the fourth annual Swirled Series, the Oakland Alphas are playing the San Francisco Gammas. The first game is played in San Francisco and succeeding games alternate in location. San Francisco has a $50\%$ chance of winning their home games, while Oakland has a probability of $60\%$ of winning at home. Normally, the series will stretch on forever until one team gets a three game lead, in which case they are declared the winners. However, after each game in San Francisco there is a $50\%$ chance of an earthquake, which will cause the series to end with the team that has won more games declared the winner. What is the probability that the Gammas will win?

Tina randomly selects two distinct numbers from the set $ \{1,2,3,4,5\}$ and Sergio randomly selects a number from the set $ \{1,2,...,10\}$. The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is $ \textbf{(A)}\ 2/5 \qquad \textbf{(B)}\ 9/20 \qquad \textbf{(C)}\ 1/2\qquad \textbf{(D)}\ 11/20 \qquad \textbf{(E)}\ 24/25$

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

Every phone number in an area consists of eight digits and starts with digit $ 8$. Mr Edy, who has just moved to the area, apply for a new phone number. What is the chance that Mr Edy gets a phone number which consists of at most five different digits?

A dart board looks like three concentric circles with radii of 4, 6, and 8. Three darts are thrown at the board so that they stick at three random locations on then board. The probability that one dart sticks in each of the three regions of the dart board is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Given is a $n \times n$ chessboard. With the same probability, we put six pawns on its six cells. Let $p_n$ denotes the probability that there exists a row or a column containing at least two pawns. Find $\lim_{n \to \infty} np_n$.

Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $ \frac{1}{6}$, independent of the outcome of any other toss.) $ \textbf{(A)}\ \frac{1}{3}\qquad \textbf{(B)}\ \frac{2}{9}\qquad \textbf{(C)}\ \frac{5}{18}\qquad \textbf{(D)}\ \frac{25}{91}\qquad \textbf{(E)}\ \frac{36}{91}$

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} $

PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, the probability that the 4th customer orders a previously unordered item is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

There is a board numbered $-10$ to $10$. Each square is coloured either red or white, and the sum of the numbers on the red squares is $n$. Maureen starts with a token on the square labeled $0$. She then tosses a fair coin ten times. Every time she flips heads, she moves the token one square to the right. Every time she flips tails, she moves the token one square to the left. At the end of the ten flips, the probability that the token finishes on a red square is a rational number of the form $\frac a b$. Given that $a + b = 2001$, determine the largest possible value for $n$.

Let $H$ be a regular hexagon with side length one. Peter picks a point $P$ uniformly and at random within $H$, then draws the largest circle with center $P$ that is contained in $H$. What is this probability that the radius of this circle is less than $1/2$ ?

Tina writes four letters to her friends Silas, Jessica, Katie, and Lekan. She prepares an envelope for Silas, an envelope for Jessica, an envelope for Katie, and an envelope for Lekan. However, she puts each letter into a random envelope. What is the probability that no one receives the letter they are supposed to receive?

$\frac{n(n + 1)}{2}$ distinct numbers are arranged at random into $n$ rows. The first row has $1$ number, the second has $2$ numbers, the third has $3$ numbers and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers.

Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords are the sides of a convex quadrilateral? $ \textbf{(A)}\ \frac{1}{15}\qquad \textbf{(B)}\ \frac{1}{91}\qquad \textbf{(C)}\ \frac{1}{273}\qquad \textbf{(D)}\ \frac{1}{455}\qquad \textbf{(E)}\ \frac{1}{1365}$

A $3\times3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black? $ \textbf{(A)}\ \dfrac{49}{512} \qquad\textbf{(B)}\ \dfrac{7}{64} \qquad\textbf{(C)}\ \dfrac{121}{1024} \qquad\textbf{(D)}\ \dfrac{81}{512} \qquad\textbf{(E)}\ \dfrac{9}{32} $

When a fair six-sided dice is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the $5$ faces than can be seen is divisible by $6$? $\textbf{(A)}\ 1/3 \qquad \textbf{(B)}\ 1/2 \qquad \textbf{(C)}\ 2/3 \qquad \textbf{(D)}\ 5/6 \qquad \textbf{(E)}\ 1$

Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does not exceed 1 percent.