Title is self explanatory. Pick two points on the unit sphere. What is the expected distance between them?

Bernardo randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a $ 3$-digit number. Silvia randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a $ 3$-digit number. What is the probability that Bernardo's number is larger than Silvia's number? $ \textbf{(A)}\ \frac {47}{72}\qquad \textbf{(B)}\ \frac {37}{56}\qquad \textbf{(C)}\ \frac {2}{3}\qquad \textbf{(D)}\ \frac {49}{72}\qquad \textbf{(E)}\ \frac {39}{56}$

Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Real numbers $X_1, X_2, \dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$. If the expected value of $\min(X_1,X_2,\dots, X_{10})^4$ can be expressed as a rational number $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$? [i]2016 CCA Math Bonanza Lightning #4.4[/i]

Let $n$ be a positive integer. Let $(a, b, c)$ be a random ordered triple of nonnegative integers such that $a + b + c = n$, chosen uniformly at random from among all such triples. Let $M_n$ be the expected value (average value) of the largest of $a$, $b$, and $c$. As $n$ approaches infinity, what value does $\frac{M_n}{n}$ approach?

At Rachelle's school an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on the four classes she is taking is computed as the total sum of points divided by $4$. She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History. She think she has a $\frac{1}{6}$ chance of getting an A in English, and a $\frac{1}{4}$ chance of getting a B. In History, she has a $\frac{1}{4}$ chance of getting an A, and a $\frac{1}{3}$ chance of getting a B, independently of what she gets in English. What is the probability that Rachelle will get a GPA of at least 3.5? $\textbf{(A) }\frac{11}{72}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{3}{16}\qquad\textbf{(D) }\frac{11}{24}\qquad\textbf{(E) }\frac{1}{2}$

From a bag containing 5 pairs of socks, each pair a different color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are 4 socks of different colors held outside the bag. What is the probability of the latter alternative?

The Nationwide Basketball Society (NBS) has $8001$ teams, numbered $2000$ through $10000$. For each $n$, team $n$ has $n+1$ players, and in a sheer coincidence, this year each player attempted $n$ shots and on team $n$, exactly one player made $0$ shots, one player made $1$ shot, . . ., one player made $n$ shots. A player's [i]field goal percentage[/i] is defined as the percentage of shots the player made, rounded to the nearest tenth of a percent (For instance, $32.45\%$ rounds to $32.5\%$). A player in the NBS is randomly selected among those whose field goal percentage is $66.6\%$. If this player plays for team $k$, the probability that $k\geq 6000$ can be expressed as $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $p+q$.

Tamika selects two different numbers at random from the set $ \{ 8,9,10\} $ and adds them. Carlos takes two different numbers at random from the set $ \{ 3,5,6\} $ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? $ \text{(A)}\ \frac{4}{9}\qquad\text{(B)}\ \frac{5}{9}\qquad\text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{1}{3}\qquad\text{(E)}\ \frac{2}{3} $

If $a$ and $b$ are each randomly and independently chosen in the interval $[-1, 1]$, what is the probability that $|a|+|b|<1$?

A blackboard contains 68 pairs of nonzero integers. Suppose that for each positive integer $k$ at most one of the pairs $(k, k)$ and $(-k, -k)$ is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student then scores one point for each of the 68 pairs in which at least one integer is erased. Determine, with proof, the largest number $N$ of points that the student can guarantee to score regardless of which 68 pairs have been written on the board.

A point $ P$ is performing a random walk on the $ X$-axis. At the instant $ t\equal{}0$, $ P$ is at a point $ x_0$ ($ |x_0|\le N$, where $ x_0$ and $ N$ denote integers, $ N>0$). If at an instant $ t$ ($ t$ being a nonnegative integer), $ P$ is at a point of $ x$ integer abscissa and $ |x|<N$, then by the instant $ t\plus{}1$ it reaches either the point $ x\plus{}1$ or the point $ x\minus{}1$, each with probability $ \frac12$. If at the instant $ t$, $ P$ is at the point $ x\equal{}N$ [$ x\equal{}\minus{}N$], then by the instant $ t\plus{}1$ it is certain to reach the point $ N\minus{}1$ [$ \minus{}N\plus{}1$]. Denote by $ P_k(t)$ the probability of $ P$ being at $ x\equal{}k$ at instant $ t$ ($ k$ is an integer). Find $ \lim_{t\to \infty}P_{k}(2t)$ and $ \lim_{t\to \infty}P_k(2t\plus{}1)$ for every fixed $ k$.

Ten balls numbered $1$ to $10$ are in a jar. Jack reaches into the jar and randomly removes one of the balls. Then Jill reaches into the jar and randomly removes a different ball. The probability that the sum of the two numbers on the balls removed is even is $\text{(A)}\ \frac{4}{9} \qquad \text{(B)}\ \frac{9}{19} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{10}{19} \qquad \text{(E)}\ \frac{5}{9}$

One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is $ \frac23$ of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is $ \textbf{(A)}\ \frac13 \qquad \textbf{(B)}\ \frac25 \qquad \textbf{(C)}\ \frac12 \qquad \textbf{(D)}\ \frac35 \qquad \textbf{(E)}\ \frac23$

Let $n$ be a positive integer. Let $f(n)$ be the probability that, if divisors $a, b, c$ of $n$ are selected uniformly at random with replacement, then $\gcd(a, \text{lcm}(b, c)) = \text{lcm}(a, \gcd(b, c))$. Let $s(n)$ be the sum of the distinct prime divisors of $n$. If $f(n) < \frac{1}{2018}$, compute the smallest possible value of $s(n)$.

A particle moves from $(0, 0)$ to $(n, n)$ directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At $(n, y), y < n$, it stays there if a head comes up and at $(x, n), x < n$, it stays there if a tail comes up. Let$ k$ be a fixed positive integer. Find the probability that the particle needs exactly $2n+k$ tosses to reach $(n, n).$

Four boys and four girls each bring one gift to a Christmas gift exchange. On a sheet of paper, each boy randomly writes down the name of one girl, and each girl randomly writes down the name of one boy. At the same time, each person passes their gift to the person whose name is written on their sheet. Determine the probability that [i]both[/i] of these events occur: [list] [*] (i) Each person receives exactly one gift; [*] (ii) No two people exchanged presents with each other (i.e., if $A$ gave his gift to $B$, then $B$ did not give her gift to $A$).[/list]

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}$

Suppose that n numbers $x_1, x_2, . . . , x_n$ are chosen randomly from the set $\{1, 2, 3, 4, 5\}$. Prove that the probability that $x_1^2+ x_2^2 +\cdots+ x_n^2 \equiv 0 \pmod 5$ is at least $\frac 15.$

On an infinite chessboard (whose squares are labeled by $ (x, y)$, where $ x$ and $ y$ range over all integers), a king is placed at $ (0, 0)$. On each turn, it has probability of $ 0.1$ of moving to each of the four edge-neighboring squares, and a probability of $ 0.05$ of moving to each of the four diagonally-neighboring squares, and a probability of $ 0.4$ of not moving. After $ 2008$ turns, determine the probability that the king is on a square with both coordinates even. An exact answer is required.

Fix positive integers $n$ and $k$ such that $2 \le k \le n$ and a set $M$ consisting of $n$ fruits. A [i]permutation[/i] is a sequence $x=(x_1,x_2,\ldots,x_n)$ such that $\{x_1,\ldots,x_n\}=M$. Ivan [i]prefers[/i] some (at least one) of these permutations. He realized that for every preferred permutation $x$, there exist $k$ indices $i_1 < i_2 < \ldots < i_k$ with the following property: for every $1 \le j < k$, if he swaps $x_{i_j}$ and $x_{i_{j+1}}$, he obtains another preferred permutation. \\ Prove that he prefers at least $k!$ permutations.

Sonya the frog chooses a point uniformly at random lying within the square $[0, 6] \times [0, 6]$ in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from $[0, 1]$ and a direction uniformly at random from {north, south east, west}. All he choices are independent. She now hops the distance in the chosen direction. What is the probability that she lands outside the square? $\textbf{(A) } \frac{1}{6} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{4} \qquad \textbf{(D) } \frac{1}{10} \qquad \textbf{(E) } \frac{1}{9}$

A given sequence $r_1, r_2, \dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more ``bubble passes''. A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the third term with the second term and exchanging them if and only if the third term is smaller, and so on in order, through comparing the last term, $r_n$, with its current predecessor and exchanging them if and only if the last term is smaller. The example below shows how the sequence 1, 9, 8, 7 is transformed into the sequence 1, 8, 7, 9 by one bubble pass. The numbers compared at each step are underlined. \[ \begin{array}{c} \underline{1 \quad 9} \quad 8 \quad 7 \\ 1 \quad \underline{9 \quad 8} \quad 7 \\ 1 \quad 8 \quad \underline{9 \quad 7} \\ 1 \quad 8 \quad 7 \quad 9 \end{array} \] Suppose that $n = 40$, and that the terms of the initial sequence $r_1, r_2, \dots, r_{40}$ are distinct from one another and are in random order. Let $p/q$, in lowest terms, be the probability that the number that begins as $r_{20}$ will end up, after one bubble pass, in the $30^{\text{th}}$ place. Find $p + q$.

During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a 100 foot by 100 foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as \[\dfrac{a-b\sqrt c}d,\] where all four variables are positive integers, $c$ is a multple of no perfect square greater than $1$, $a$ is coprime with $d$, and $b$ is coprime with $d$. Find the value of $a+b+c+d$.

$32$ 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 $m+n$.