You drop a 7 cm long piece of mechanical pencil lead on the floor. A bully takes the lead and breaks it at a random point into two pieces. A piece of lead is unusable if it is 2 cm or shorter. If the expected value of the number of usable pieces afterwards is $\frac{m}n$ for relatively prime positive integers $m$ and $n$, compute $100m + n$. [i]Proposed by Aaron Lin[/i]

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 $m+n$.

$16$ white and $4$ red balls that are not identical are distributed randomly into $4$ boxes which contain at most $5$ balls. What is the probability that each box contains exactly $1$ red ball? $ \textbf{(A)}\ \dfrac{5}{64} \qquad\textbf{(B)}\ \dfrac{1}{8} \qquad\textbf{(C)}\ \dfrac{4^4}{\binom{16}{4}} \qquad\textbf{(D)}\ \dfrac{5^4}{\binom{20}{4}} \qquad\textbf{(E)}\ \dfrac{3}{32} $

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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}$

A frog is jumping on $N$ stones which are numbered from $1$ to $N$ from left to right. The frog is jumping to the previous stone (to the left) with probability $p$ and is jumping to the next stone (to the right) with probability $1-p$. If the frog has jumped to the left from the leftmost stone or to the right from the rightmost stone, it will fall into the water. The frog is initially on the leftmost stone. If $p< \tfrac 13$, show that the frog will fall into the water from the rightmost stone with a probability higher than $\tfrac 12$.

Select a random real number $m$ from the interval $(\frac{1}{6}, 1)$. A track is in the shape of an equilateral triangle of side length $50$ feet. Ch, Hm, and Mc are all initially standing at one of the vertices of the track. At the time $t = 0$, the three people simultaneously begin walking around the track in clockwise direction. Ch, Hm, and Mc walk at constant rates of $2, 3$, and $4$ feet per second, respectively. Let $T$ be the set of all positive real numbers $t_0$ satisfying the following criterion: [i]If we choose a random number $t_1$ from the interval $[0, t_0]$, the probability that the three people are on the same side of the track at the time $t = t_1$ is precisely $m$.[/i] The probability that $|T| = 17$ (i.e., $T$ has precisely $17$ elements) equals $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Alice, Bob, and Chris each roll $4$ dice. Each only knows the result of their own roll. Alice claims that there are at least $5$ multiples of $3$ among the dice rolled. Bob has $1$ six and no threes, and knows that Alice wouldn’t claim such a thing unless he had at least $2$ multiples of $3$. Bob can call Alice a liar, or claim that there are at least $6$ multiples of $3$, but Chris says that he will immediately call Bob a liar if he makes this claim. Bob wins if he calls Alice a liar and there aren't at least $5$ multiples of $3$, or if he claims there are at least $6$ multiples of $3$, and there are. What is the probability that Bob loses no matter what he does?

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.

The $ 52$ cards in a deck are numbered $ 1, 2, \ldots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let $ p(a)$ be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards $ a$ and $ a\plus{}9$, and Dylan picks the other of these two cards. The minimum value of $ p(a)$ for which $ p(a)\ge\frac12$ can be written as $ \frac{m}{n}$. where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

Let $n$ and $k$ be positive integers with $1 \leq k \leq n$. A set of cards numbered $1$ to $n$ are arranged randomly in a row from left to right. A person alternates between performing the following moves: [list=a] [*] The leftmost card in the row is moved $k-1$ positions to the right while the cards in positions $2$ through $k$ are each moved one place to the left. [*] The rightmost card in the row is moved $k-1$ positions to the left while the cards in positions $n-k+1$ through $n-1$ are each moved one place to the right. [/list] Determine the probability that after some number of moves the cards end up in order from $1$ to $n$, left to right.

If 100 dice are rolled, what is the probability that the sum of the numbers rolled is even? [i]Lightning 1.1[/i]

Kailey starts with the number $0$, and she has a fair coin with sides labeled $1$ and $2$. She repeatedly flips the coin, and adds the result to her number. She stops when her number is a positive perfect square. What is the expected value of Kailey’s number when she stops? If E is your estimate and A is the correct answer, you will receive $\left\lfloor 25e^{-\frac{5|E-A|}{2} }\right\rfloor$ points.

Two different numbers are randomly selected from the set ${ - 2, -1, 0, 3, 4, 5}$ and multiplied together. What is the probability that the product is $0$? $\textbf{(A) }\dfrac{1}{6}\qquad\textbf{(B) }\dfrac{1}{5}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad \textbf{(E) }\dfrac{1}{2}$

Let $X_1, X_2$ be independent, identically distributed random variables such that $X_i\geq 0$ for all $i$. Let $\mathrm EX_i=m$, $\mathrm{Var} (X_i)=\sigma ^2<\infty$. Show that, for all $0<\alpha\leq 1$ $$\lim_{n\to\infty} n\,\mathrm{Var} \left( \left[ \frac{X_1+\ldots +X_n}{n}\right] ^\alpha\right)=\frac{\alpha ^ 2 \sigma ^ 2}{m^{2(1-\alpha)}}$$ [Gy. Michaletzki]

Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

A jury of $9$ persons should decide whether a verdict is guilty or not. Each juror votes independently with the probability $1/2$ for each of the two possibilities, and noone is allowed to be abstinent. Find the probability that a fixed juror will be a part of the majority. In the case of a jury of $n$ persons, find the values of n for which the probability of being a part of the majority is greater than, equal to, and smaller than $1/2$, respectively. (For $n = 2k$, $k +1$ votes are needed for a majority.)

Two circles of radius 1 are to be constructed as follows. The center of circle $ A$ is chosen uniformly and at random from the line segment joining $ (0,0)$ and $ (2,0)$. The center of circle $ B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $ (0,1)$ to $ (2,1)$. What is the probability that circles $ A$ and $ B$ intersect? $ \textbf{(A)} \; \frac{2\plus{}\sqrt{2}}{4} \qquad \textbf{(B)} \; \frac{3\sqrt{3}\plus{}2}{8} \qquad \textbf{(C)} \; \frac{2 \sqrt{2} \minus{} 1}{2} \qquad \textbf{(D)} \; \frac{2\plus{}\sqrt{3}}{4} \qquad \textbf{(E)} \; \frac{4 \sqrt{3} \minus{} 3}{4}$

We roll a regular 6-sided dice $n$ times. What is the probabilty that the total number of eyes rolled is a multiple of 5?

Jackie and Phil have two fair coins and a third coin that comes up heads with probability $ \frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $ \frac{m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

Consider a fair coin and a fair 6-sided die. The die begins with the number 1 face up. A [i]step[/i] starts with a toss of the coin: if the coin comes out heads, we roll the die; otherwise (if the coin comes out tails), we do nothing else in this step. After 5 such steps, what is the probability that the number 1 is face up on the die?

The Miami Heat and the San Antonio Spurs are playing a best-of-five series basketball championship, in which the team that first wins three games wins the whole series. Assume that the probability that the Heat wins a given game is $x$ (there are no ties). The expected value for the total number of games played can be written as $f(x)$, with $f$ a polynomial. Find $f(-1)$.

A collection of $2n$ letters contains $2$ each of $n$ different letters. The collection is partitioned into $n$ pairs, each pair containing $2$ letters, which may be the same or different. Denote the number of distinct partitions by $u_n$. (Partitions differing in the order of the pairs in the partition or in the order of the two letters in the pairs are not considered distinct.) Prove that $u_{n+1}=(n+1)u_n - \frac{n(n-1)}{2} u_{n-2}.$ [i]Similar Problem :[/i] A pack of $2n$ cards contains $n$ pairs of $2$ identical cards. It is shuffled and $2$ cards are dealt to each of $n$ different players. Let $p_n$ be the probability that every one of the $n$ players is dealt two identical cards. Prove that $\frac{1}{p_{n+1}}=\frac{n+1}{p_n} + \frac{n(n-1)}{2p_{n-2}}.$

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 $p+q$.

Let $S$ be a randomly chosen $6$-element subset of the set $\{0,1,2,\ldots,n\}.$ Consider the polynomial $P(x)=\sum_{i\in S}x^i.$ Let $X_n$ be the probability that $P(x)$ is divisible by some nonconstant polynomial $Q(x)$ of degree at most $3$ with integer coefficients satisfying $Q(0) \neq 0.$ Find the limit of $X_n$ as $n$ goes to infinity.