M203 20260516 Counting and Probability (Part 3)
2008 AMC 12B Problems/Problem 22
A parking lot has $16$ spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires $2$ adjacent spaces. What is the probability that she is able to park?
$\textbf{(A)}\ \frac{11}{20}\qquad \textbf{(B)}\ \frac47\qquad \textbf{(C)}\ \frac{81}{140}\qquad \textbf{(D)}\ \frac35\qquad \textbf{(E)}\ \frac{17}{28}$
#Binomial_Distribution
For $n$ independent trials with probability of success $p$, the probability of exactly $m$ successes is
$$P(X=m)=\binom{n}{m}p^m(1-p)^{n-m}.$$
2011 AMC 12A Problems/Problem 14
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 $y=ax^2-bx$?
$\textbf{(A)}\ \frac{11}{81} \qquad \textbf{(B)}\ \frac{13}{81} \qquad \textbf{(C)}\ \frac{5}{27} \qquad \textbf{(D)}\ \frac{17}{81} \qquad \textbf{(E)}\ \frac{19}{81}$
2012 AMC 10B Problems/Problem 24
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?
$\textbf{(A)}\ 108 \qquad \textbf{(B)}\ 132 \qquad \textbf{(C)}\ 671 \qquad \textbf{(D)}\ 846 \qquad \textbf{(E)}\ 1105$
2015 AMC 10A Problems/Problem 25
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 $\frac{1}{2}$ is $\frac{a-b\pi}{c}$, where $a$, $b$, and $c$ are positive integers with $\gcd(a,b,c)=1$. What is $a+b+c$?
$\textbf{(A)}\ 59\qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 61\qquad \textbf{(D)}\ 62\qquad \textbf{(E)}\ 63$
2016 AMC 10A Problems/Problem 17
Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\frac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\frac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N)<\frac{321}{400}$?
$\textbf{(A)}\ 12\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 20$
2017 AMC 10A Problems/Problem 15
Chloe chooses a real number uniformly at random from the interval $[0,2017]$. Independently, Laurent chooses a real number uniformly at random from the interval $[0,4034]$. What is the probability that Laurent's number is greater than Chloe's number?
$\textbf{(A)}\ \frac{1}{2}\qquad \textbf{(B)}\ \frac{2}{3}\qquad \textbf{(C)}\ \frac{3}{4}\qquad \textbf{(D)}\ \frac{5}{6}\qquad \textbf{(E)}\ \frac{7}{8}$
#Geometric_Probability
2017 AMC 10A Problems/Problem 18
Amelia has a coin that lands heads with probability $\frac{1}{3}$, and Blaine has a coin that lands on heads with probability $\frac{2}{5}$. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $q-p$?
$\textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 5$
2018 AMC 10B Problems/Problem 22
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[0,1]$. Which of the following numbers is closest to the probability that $x,y,$ and $1$ are the side lengths of an obtuse triangle?
$\textbf{(A)} \text{ 0.21} \qquad \textbf{(B)} \text{ 0.25} \qquad \textbf{(C)} \text{ 0.29} \qquad \textbf{(D)} \text{ 0.50} \qquad \textbf{(E)} \text{ 0.79}$
2018 AMC 10A Problems/Problem 19
A number $m$ is randomly selected from the set $\{11,13,15,17,19\}$, and a number $n$ is randomly selected from $\{1999,2000,2001,\ldots,2018\}$. What is the probability that $m^n$ has a units digit of $1$?
$\textbf{(A)}\ \frac{1}{5}\qquad \textbf{(B)}\ \frac{1}{4}\qquad \textbf{(C)}\ \frac{3}{10}\qquad \textbf{(D)}\ \frac{7}{20}\qquad \textbf{(E)}\ \frac{2}{5}$
1:38:38（Homework Problem）
2015 AMC 10B Problems/Problem 16
Al, Bill, and Cal will each randomly be assigned a whole number from $1$ to $10$, inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's?
$\textbf{(A)}\ \dfrac{9}{1000}\qquad \textbf{(B)}\ \dfrac{1}{90}\qquad \textbf{(C)}\ \dfrac{1}{80}\qquad \textbf{(D)}\ \dfrac{1}{72}\qquad \textbf{(E)}\ \dfrac{2}{121}$
Purple Comet 2026 MS - Problems 19 and 20
Problem 19
Five red plates, five white plates, and five blue plates are placed in a circle in random order. The probability that no two red plates are adjacent is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Teacher's favorite: https://www.wolframalpha.com/
Charley is building an ice cream sundae at Isabella's Impressive Ice Cream. There are 4 different flavors of ice cream and $7$ different toppings from which he can choose. He can choose any whole number of toppings and ice cream flavors, but he must have at least one flavor of ice cream and at at least one topping. Given that the order in which he chooses the ice cream flavors and the toppings does not matter, how many different sundaes can Charley make?
Charley the Ice Cream Connoisseur
Berkeley Math Tournament 2025 Discrete Test
Problem 9
In a tournament, twelve players compete in a sequence of matches. Each match consists of two players, and no two matches have the same pair of players. Furthermore, no two players who compete in the same number of matches in the tournament compete against each other in a match. What is the greatest possible number of matches that can be played in the tournament?