Olympiad problems

1986 AMC 8, 24

Tags: probability

The $ 600$ students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of the three lunch groups. The probability that the three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately: \[ \textbf{(A)}\ \frac{1}{27} \qquad \textbf{(B)}\ \frac{1}{9} \qquad \textbf{(C)}\ \frac{1}{8} \qquad \textbf{(D)}\ \frac{1}{6} \qquad \textbf{(E)}\ \frac{1}{3} \]

2019 Harvard-MIT Mathematics Tournament, 4

Tags: hmmt , probability , expected value

Yannick is playing a game with $100$ rounds, starting with $1$ coin. During each round, there is an $n\%$ chance that he gains an extra coin, where $n$ is the number of coins he has at the beginning of the round. What is the expected number of coins he will have at the end of the game?

2001 Stanford Mathematics Tournament, 11

Tags: college , probability

Christopher and Robin are playing a game in which they take turns tossing a circular token of diameter 1 inch onto an inﬁnite checkerboard whose squares have sides of 2 inches. If the token lands entirely in a square, the player who tossed the token gets 1 point; otherwise, the other player gets 1 point. A player wins as soon as he gets two more points than the other player. If Christopher tosses ﬁrst, what is the probability that he will win? Express your answer as a fraction.

2021 AMC 10 Fall, 14

Tags: probability

Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4?$ $\textbf{(A)}\: \frac34\qquad\textbf{(B)} \: \frac{57}{64}\qquad\textbf{(C)} \: \frac{59}{64}\qquad\textbf{(D)} \: \frac{187}{192}\qquad\textbf{(E)} \: \frac{63}{64}$

2002 AIME Problems, 12

Tags: probability , ratio

A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_{n}$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10}=.4$ and $a_{n}\le .4$ for all $n$ such that $1\le n \le 9$ is given to be $p^{a}q^{b}r/(s^{c}),$ where $p,$ $q,$ $r,$ and $s$ are primes, and $a,$ $b,$ and $c$ are positive integers. Find $(p+q+r+s)(a+b+c).$

2023 AIME, 6

Tags: probability , expected value , number theory , relatively prime

Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2022 AMC 10, 12

Tags: probability , dice

A pair of fair $6$-sided dice is rolled $n$ times. What is the least value of $n$ such that the probability that the sum of the numbers face up on a roll equals $7$ at least once is greater than $\frac{1}{2}$? $\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6$

2014 SDMO (Middle School), 2

Tags: probability , expected value

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?

2009 IMS, 6

Tags: probability , symmetry , probability and stats

Suppose that there are 100 seats in a saloon for 100 students. All students except one know their seat. First student (which is the one who doesn't know his seat) comes to the saloon and sits randomly somewhere. Then others enter the saloon one by one. Every student that enters the saloon and finds his seat vacant, sits there and if he finds his seat occupied he sits somewhere else randomly. Find the probability that last two students sit on their seats.

2009 Math Prize For Girls Problems, 12

Tags: probability , expected value

Jenny places 100 pennies on a table, 30 showing heads and 70 showing tails. She chooses 40 of the pennies at random (all different) and turns them over. That is, if a chosen penny was showing heads, she turns it to show tails; if a chosen penny was showing tails, she turns it to show heads. At the end, what is the expected number of pennies showing heads?

2006 Stanford Mathematics Tournament, 2

Tags: probability

A customer enters a supermarket. The probability that the customer buys bread is .60, the probability that the customer buys milk is .50, and the probability that the customer buys both bread and milk is .30. What is the probability that the customer would buy either bread or milk or both?

2007 AMC 8, 21

Tags: probability

Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair? $\textbf{(A)}\ \frac{2}{7} \qquad \textbf{(B)}\ \frac{3}{8} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{4}{7} \qquad \textbf{(E)}\ \frac{5}{8}$

2013 Stanford Mathematics Tournament, 3

Tags: probability , expected value

Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$. What is the expected value of the number of games they will play?

1982 IMO Shortlist, 10

Tags: probability , combinatorics , counting

A box contains $p$ white balls and $q$ black balls. Beside the box there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have different colors, the white ball is put back into the box. This procedure is repeated until the last two balls are removed from the box and one last ball is put in. What is the probability that this last ball is white?

2013 Harvard-MIT Mathematics Tournament, 15

Tags: hmmt , probability , expected value

Tim and Allen are playing a match of [i]tenus[/i]. In a match of [i]tenus[/i], the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability $3/4$, and in the even-numbered games, Allen wins with probability $3/4$. What is the expected number of games in a match?

1993 AMC 12/AHSME, 24

Tags: probability , function , complementary counting

A box contains $3$ shiny pennies and $4$ dull pennies. One by one, pennies are drawn at random from the box and not replaced. If the probability is $\frac{a}{b}$ that it will take more than four draws until the third shiny penny appears and $\frac{a}{b}$ is in lowest terms, then $a+b=$ $ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 35 \qquad\textbf{(D)}\ 58 \qquad\textbf{(E)}\ 66 $

2010 Slovenia National Olympiad, 4

Tags: probability , function , inequalities

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?

2006 Pre-Preparation Course Examination, 1

Tags: limit , vector , probability , invariant , topology , real analysis , advanced fields

Suppose that $X$ is a compact metric space and $T: X\rightarrow X$ is a continous function. Prove that $T$ has a returning point. It means there is a strictly increasing sequence $n_i$ such that $\lim_{k\rightarrow \infty} T^{n_k}(x_0)=x_0$ for some $x_0$.

2007 Putnam, 6

Tags: integration , probability , analytic geometry , logarithm

For each positive integer $ n,$ let $ f(n)$ be the number of ways to make $ n!$ cents using an unordered collection of coins, each worth $ k!$ cents for some $ k,\ 1\le k\le n.$ Prove that for some constant $ C,$ independent of $ n,$ \[ n^{n^2/2\minus{}Cn}e^{\minus{}n^2/4}\le f(n)\le n^{n^2/2\plus{}Cn}e^{\minus{}n^2/4}.\]

2001 Stanford Mathematics Tournament, 8

Tags: college , probability

Janet and Donald agree to meet for lunch between 11:30 and 12:30. They each arrive at a random time in that interval. If Janet has to wait more than 15 minutes for Donald, she gets bored and leaves. Donald is busier so will only wait 5 minutes for Janet. What is the probability that the two will eat together? Express your answer as a fraction.

2001 Finnish National High School Mathematics Competition, 4

Tags: probability , combinatorics

A sequence of seven digits is randomly chosen in a weekly lottery. Every digit can be any of the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$ What is the probability of having at most five different digits in the sequence?

2005 Today's Calculation Of Integral, 34

Tags: calculus , integration , probability , calculus computations

Let $p$ be a constant number such that $0<p<1$. Evaluate \[\sum_{k=0}^{2004} \frac{p^k (1-p)^{2004-k}}{\displaystyle \int_0^1 x^k (1-x)^{2004-k} dx}\]

2006 AMC 10, 21

Tags: ratio , probability , function , analytic geometry

For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5 and 6 on each die are in the ratio $ 1: 2: 3: 4: 5: 6$. What is the probability of rolling a total of 7 on the two dice? $ \textbf{(A) } \frac 4{63} \qquad \textbf{(B) } \frac 18 \qquad \textbf{(C) } \frac 8{63} \qquad \textbf{(D) } \frac 16 \qquad \textbf{(E) } \frac 27$

2015 Finnish National High School Mathematics Comp, 5

Tags: point , correct , probability , combinatorics

Mikko takes a multiple choice test with ten questions. His only goal is to pass the test, and this requires seven points. A correct answer is worth one point, and answering wrong results in the deduction of one point. Mikko knows for sure that he knows the correct answer in the six first questions. For the rest, he estimates that he can give the correct answer to each problem with probability $p, 0 < p < 1$. How many questions Mikko should try?

2011 AMC 12/AHSME, 14

Tags: probability , parabola , inequalities , conic

Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. What is the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$? $ \textbf{(A)}\ \frac{11}{81} \qquad \textbf{(B)}\ \frac{13}{81} \qquad \textbf{(C)}\ \frac{5}{27} \qquad \textbf{(D)}\ \frac{17}{81} \qquad \textbf{(E)}\ \frac{19}{81} $