Probability - Class 11

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8\}$ and $A = (1, 6) \cap \mathbb{Z}$. Find the total number of bijective functions $f: S \to S$ satisfying the following two conditions:

1. $f(x) \neq x$ for all $x \in S$.
2. The image set $f(A)$ is closed under the binary operation $a \oplus b = |a - b| + 1$.

Let $X$ be a continuous random variable with probability density function $f(x) = k(x^2 + |x|)$ for $x \in [-3, 3]$ and $f(x) = 0$ otherwise. Let $E$ be the event that the random variable $X$ satisfies the inequality: $\int_{-X}^{X} \left( t^3 \cos t + t^2 \sin^3 t + |t| - 1 \right) dt > 0$ Find the probability of event $E$, $P(E)$.

Let $f(x) = (\sin 2x)^{\sin 2x}$ be defined on its maximal domain $D \subset (0, 2\pi)$ where the base is strictly positive. Let $A = \{x \in D : f'(x) = 0\}$ and $B = \{x \in (0, 2\pi) \setminus D : x \text{ is a limit point of } D \text{ and } \lim_{t \to x, t \in D} f(t) \text{ exists finitely}\}$. A sample space is defined as the set of discrete points $S = A \cup B$. If three distinct elements are chosen at random from $S$, what is the probability that exactly one of them belongs to $A$?

Let $S$ be the set of all ordered triplets $(x,y,z)$ of non-negative integers such that $x+y+z \le 20$. The sample space $\Omega$ is defined as the subset of $S$ for which the expression $\frac{1}{x-y}$ is a well-defined real number. If a triplet is chosen uniformly at random from $\Omega$, what is the probability that $z > x + y$?

In a geometric probability model, the probability of an event $E$ is given by the formula $P(E) = \frac{1}{10} [\vec{u}\ \vec{v}\ \vec{w}]$, where $[\vec{u}\ \vec{v}\ \vec{w}]$ denotes the scalar triple product $\vec{u} \cdot (\vec{v} \times \vec{w})$. Let $\vec{u} = x\hat{i} + 3\hat{j} - \hat{k}$, $\vec{v} = \hat{i} - 2\hat{j} + \hat{k}$, and $\vec{w} = 2\hat{i} + x\hat{j} + 2\hat{k}$. If $x$ is an integer, what is the sum of all possible values of $x$ for which $P(E)$ satisfies the axioms of probability?

A continuous random variable $X$ takes values in the interval $[0, \pi]$. Its probability density function is given by $f(x) = K(F''(x) + F(x))$, where $K$ is a positive constant and $F(x) = \int_{0}^{x} \sin(x-t) \arccos(\cos 2t) dt$ What is the variance of $X$?

Let $A$ and $B$ be two events associated with a random experiment. The probability of event $A$ is given by $P(A) = \det(M)$, where $M$ is the $3 \times 3$ matrix: $M = \begin{pmatrix} \int_{-\pi/4}^{\pi/4} \ln\left(\frac{1-\sin x}{1+\sin x}\right) dx & \alpha^2+1 & \beta \\ -(\alpha^2+1) & \sum_{k=1}^{59} \left( \cos\frac{k\pi}{2} - \cos\frac{-k\pi}{2} \right) & \gamma \\ -\beta & -\gamma & \lim_{x \to 0} \frac{e^{x^2} - 1 - x^2}{x^4} - \frac{1}{2} \end{pmatrix}$ for some real constants $\alpha, \beta, \gamma$. Event $B$ is defined as the event that a number $t$ chosen at random from the sample space $S = \{t \in \mathbb{Z} : -59 \le t \le 283\}$ satisfies the inequality: $(t-59)^2 (t-283) < 0$ Find the value of $343 \times P(A^c \cap B)$.

Let $M = \begin{bmatrix} 1 & -1 & -1 \\ 0 & 2 & 1 \\ 0 & 0 & 1 \end{bmatrix}$. Let $\Omega$ be the region in $\mathbb{R}^3$ defined by the system of inequalities: $x \ge 0, y \ge 0, z \ge 0$, $x + y + z \le 6$, and $2x + y + z \le 13$. A point $\mathbf{v} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is chosen uniformly at random from $\Omega$. Let $E$ be the event that the vector $\mathbf{u} = \text{adj}(M) \mathbf{v}$ also lies in $\Omega$. Find the probability of the event $E$.

A biased coin with probability of showing heads $p \in (0, 1)$ is tossed $6$ times independently. Let $A$ be the event that the first toss shows heads, and $S$ be the event that the total number of heads obtained in the $6$ tosses is either $2$ or $3$. If $A$ and $S$ are independent events, then the value of $p$ is:

Let $A, B, C$ be three events in a probability space such that $A$ and $C$ are mutually exclusive. The events $A$ and $B$ are independent, and $B$ and $C$ are independent. Given that $P(A \cup B) = \frac{3}{4}$, $P(B \cup C) = \frac{5}{8}$, and $P(A \cup C) = \frac{3}{4}$. Let $p_1$ be the probability that exactly one of the events $A, B, C$ occurs, and $p_2$ be the probability that exactly two of the events $A, B, C$ occur. The value of $\frac{p_1}{p_2}$ is:

Let f(x) = \\begin{cases} ax + b & \\text{if } x \\ge 0 \\\\ cx^2 + d & \\text{if } x < 0 \\end{cases} and $g(x) = x^2 - 1$. A 4-tuple $(a, b, c, d)$ is chosen uniformly at random from the set $S = \\{-2, -1, 0, 1, 2\\}^4$. Let $A$ be the event that the composite function $h(x) = f(g(x))$ is continuous at $x = 1$, and $B$ be the event that $h(x)$ is differentiable at $x = 1$. Which of the following statements is correct?

Let $S$ be the set of all $3 \times 3$ diagonal matrices whose principal diagonal elements are chosen from the set $\{0, 1\}$. Two matrices $A$ and $B$ are chosen randomly and independently from $S$ with replacement. Let $E$ be the event that $\det(2A^T B) = 0$ and $F$ be the event that $\det(A - B) = 0$. The probability $P(E \cap F)$ is equal to

Let $S$ be a set containing 5 distinct elements. Let $A$ be the family of all subsets of $S$ having an even number of elements, and $B$ be the family of all subsets of $S$ having an odd number of elements. A random sample of 16 subsets is chosen without replacement from the power set of $S$. Let $E$ be the event that the sample contains an even number of subsets from $A$, and $F$ be the event that the sample contains an even number of subsets from $B$. What is the probability of $E \cup F$?

Four identical balls are distributed randomly into three distinct boxes, such that each ball is independently placed into any of the boxes with equal probability. Let $P_0$ be the probability that the first box is empty, and $P_2$ be the probability that the first box contains exactly two balls. Find the sum of all integers $x \in [1, 10]$ that satisfy the inequality: $\frac{x^2 - 11x + 28}{P_0 - P_2} \ge 0$

Urn $U_1$ contains $x-2$ red balls and $5$ black balls. Urn $U_2$ contains $x-2$ red balls and $1$ black ball, where $x \ge 2$ is an integer. An urn is selected by rolling a fair standard die. If the die shows a number less than 3, Urn $U_1$ is chosen; otherwise, Urn $U_2$ is chosen. A ball is drawn at random from the selected urn and is found to be red. Given that the drawn ball is red, the probability that it came from Urn $U_1$ is $\frac{3}{10}$. Find the sum of all possible valid values of $x$.

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors such that the volume of the parallelepiped formed by them is $325$. A point $P$ with position vector $\vec{r} = x\vec{a} + y\vec{b} + z\vec{c}$ is chosen uniformly at random inside this parallelepiped. What is the volume of the region consisting of all such points $P$ for which line segments of lengths $x, y,$ and $z$ can form an acute-angled triangle?

A discrete random variable $X$ takes values in the set $S = \{-3, -2, -1, 0, 1, 2, 3\}$. The probability mass function of $X$ is given by $P(X=k) = C \left( 5 - |k| + \lambda \cdot \text{sgn}(k^3 - k) \right)$ where $C$ and $\lambda$ are real constants, and $\text{sgn}(\cdot)$ denotes the signum function. If the expected value of $X$ is $E(X) = \frac{20}{23}$, then the probability $P(X \ge 1)$ is equal to

Let $W$ be the $80^{\text{th}}$ word in the dictionary of all unique permutations of the letters $\text{A, A, B, C, D, E}$. A fair 6-sided die has its faces marked with the letters $\text{A, A, B, C, D, E}$. A 6-letter word $X$ is generated by rolling this die 6 times and recording the letters in the order they appear. What is the probability that $X$ is a permutation of the letters of $W$, and has the same starting letter and the same ending letter as $W$?

Let $A = \{1, 2, 3, 4, 5, 6\}$ and $B = \left\{ x \in \mathbb{N} : x \le 5 \text{ and } \frac{x-3}{x^2-5x+4} \in \mathbb{R} \right\}$. Find the number of surjective functions $f: A \to B$ such that $f(1) + f(2) = 7$.

Let $S = \{z \in \mathbb{C} : z^{12} = 1\}$. Find the number of matrices $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with entries from $S$ such that $A \bar{A} \neq \bar{A} A$, where $\bar{A}$ is the complex conjugate of $A$.

A light ray travels along the line $L: \frac{x}{3} = \frac{y}{4} = \frac{z}{12}$ and intersects a series of parallel glass plates. The surfaces of the plates lie on the planes $\Pi_k: 3x + 4y = 25k$ for $k = 1, 2, \dots, 40$. Let the intersection points of the ray with these planes be $P_1, P_2, \dots, P_{40}$ respectively. A subset of 7 points is to be chosen from these 40 points such that the spatial distance between the orthogonal projections of any two chosen points on the plane $\Pi: 3x + 4y = 0$ is at least 39. Find the number of ways to choose these 7 points.

Let $A$ be a $4 \times 4$ matrix defined as $A = \begin{pmatrix} x_1 & x_1 & x_1 & x_1 \\ x_1 & x_2 & x_2 & x_2 \\ x_1 & x_2 & x_3 & x_3 \\ x_1 & x_2 & x_3 & x_4 \end{pmatrix}$ where $x_1, x_2, x_3, x_4 \in \mathbb{Z}$. Find the number of distinct quadruplets $(x_1, x_2, x_3, x_4)$ such that $|\det(A)| = 293$.

Let $f(x) = \begin{cases} x^3 - 3x + k - 2 & \text{for } -2 \le x < 1 \\ ax^2 + bx + c & \text{for } 1 \le x \le 3 \end{cases}$ be a function continuous and differentiable on $(-2, 3)$, where $a, b, c$ are real constants and $f(3) = k$. Let $S$ be the set of all integers $k$ such that the equation $f(x) = 0$ has exactly 3 distinct real roots. For each $k \in S$, a bag $B_k$ is prepared containing $N_k$ red balls and $15 - N_k$ black balls, where $N_k$ is the number of non-negative integer solutions to the equation $x + y + z = k$. A bag is selected at random (all bags in $S$ are equally likely to be chosen), and two balls are drawn from it without replacement. If both balls are red, let $P$ be the probability that they were drawn from bag $B_3$. Find the value of $P$.

Let $S = \{z \in \mathbb{C} : |z| \le \sqrt{5} \text{ and } \text{Re}(z), \text{Im}(z) \in \mathbb{Z}\}$. Let $V = \{z \in S : \text{Re}(z) \cdot \text{Im}(z) = 0\}$ and $C = S \setminus V$. If all elements of $S$ are arranged randomly in a row to form a sequence, what is the probability that at least two elements of $V$ are adjacent?

Let $A, B,$ and $C$ be three independent events. A matrix $M$ is defined as: $M = \begin{bmatrix} 1 & P(A) & P(B) & P(C) \\ 1 & 1 & P(B) & P(C) \\ 1 & P(A) & 1 & P(C) \\ 1 & P(A) & P(B) & 1 \end{bmatrix}$ Let $N$ be the matrix obtained by swapping the 2nd and 3rd columns of $M$. If $3\det(N) + 5P(A' \cap B' \cap C') = \frac{1}{29}$, find the probability that at least one of the events $A, B,$ or $C$ occurs.

Let $\Omega$ be the region in the first octant bounded by the coordinate planes and the tangent plane to the surface $xyz = 8$ at the point on the surface closest to the origin. A point $(x,y,z)$ is chosen uniformly at random from $\Omega$. Let $f(t) = |t-x| + |t-y| + |t-z|$. If $E$ is the event that the absolute minimum value of $f(t)$ is at most $4$, find the probability of $E$.

Let $N$ be the exponent of the highest power of $3$ dividing $100!$, and $k$ be the exponent of the highest power of $5$ dividing $100!$. A subset $A$ is chosen at random from the power set of $S = \{1, 2, \dots, N\}$. The probability that the number of elements in $A$ is strictly greater than $k$ is $p$. Then $p$ is equal to:

Let $y = f(x)$ be the continuous solution to the differential equation $\frac{dy}{dx} + |x-2| y = |x-2|$ satisfying $f(2) = 2$. Let $k$ be the sum of the $x$-coordinates of all points of inflection of the curve $y = f(x)$. Three fair six-sided dice are rolled, and their outcomes are used as the coordinates of a point $P(a,b,c)$ in 3D space. Let $\Pi$ be the plane $2x + 2y + z = 3k$. Find the probability that the perpendicular distance from $P$ to the plane $\Pi$ is strictly greater than 1.

Let $S = \{1, 2, 3, \dots, 20\}$. A number $n$ is chosen at random from $S$. Three bags $B_1, B_2,$ and $B_3$ are available. Bag $B_1$ is selected if $|n - 8| + |n - 14| = 6$. Bag $B_2$ is selected if $|n - 8| + |n - 14| > 6$ and $|n - 11| \le 5$. Bag $B_3$ is selected if $|n - 11| > 5$. The contents of the bags are as follows: $B_1$ contains 2 Red and 3 Black balls, $B_2$ contains 3 Red and 2 Black balls, and $B_3$ contains 4 Red and 1 Black ball. A ball is drawn at random from the selected bag and is found to be Red. What is the probability that the chosen number $n$ is a prime number?

Six distinct balls are distributed randomly into four distinct boxes. Let $E$ be the event that no box is empty, and $F$ be the event that Box 1 and Box 2 contain an equal number of balls. The conditional probability $P(F|E)$ is:

A system has two states, $S_1$ and $S_2$. It can undergo two types of transition protocols, $X$ and $Y$. Under protocol $X$, a system in $S_1$ transitions to $S_2$ with probability $\frac{1}{2}$ and remains in $S_1$ with probability $\frac{1}{2}$. A system in $S_2$ remains in $S_2$ with probability $1$. Under protocol $Y$, a system in $S_1$ remains in $S_1$ with probability $1$. A system in $S_2$ transitions to $S_1$ with probability $\frac{1}{2}$ and remains in $S_2$ with probability $\frac{1}{2}$. A fair coin is tossed. If it shows Heads, the system undergoes the sequence of protocols $X, Y, X, Y$ in that order. If it shows Tails, the system undergoes the sequence $X, X, Y, Y$ in that order. Given that the system initially starts in state $S_1$ and is found to be in state $S_1$ after the four protocols are completed, what is the probability that the coin showed Heads?

A unit vector $\hat{u}$ is chosen uniformly at random in $\mathbb{R}^3$. Let $\pi$ be a plane passing through the origin with a fixed normal vector $\hat{n}$. Let $\vec{v}$ be the orthogonal projection of $\hat{u}$ onto the plane $\pi$. Consider the triangle $T$ whose vertices are the origin $O$, and the points $U$ and $V$ such that $\vec{OU} = \hat{u}$ and $\vec{OV} = \vec{v}$. Let $E_1$ be the event that the area of triangle $T$ is at most $\frac{\sqrt{3}}{8}$, and $E_2$ be the event that $|\vec{v}| \le \frac{1}{2}$. Find the conditional probability $P(E_2 \mid E_1)$.

A real number $x$ is chosen uniformly at random from the interval $[-4, 4]$. Let $A$ be the event that $x$ satisfies the inequality $\frac{x^3 - 2x^2 - 4x + 8}{x+1} \ge 0$. Let $B$ be the event that $\lim_{n \to \infty} \left( \frac{n^2 + 3xn + 1}{n^2 + xn + 1} \right)^n \ge e^2$. Find the conditional probability $P(B \mid A)$.

A system undergoes three sequential state transitions from state 0 to state 3. For a parameter $t \in (0, 1)$, the probability of transitioning from state 0 to state 1 is $t$, from state 1 to state 2 is $\sqrt{t}$, and from state 2 to state 3 is $t^{3/2}$. Let $P(t)$ be the probability that the system successfully reaches state 3 from state 0.

Three points in 3D space have position vectors: $\vec{a} = (\cos^{-1}t) \hat{i} + t \hat{j}$ $\vec{b} = (\cos^{-1}t) \hat{j} + \hat{k}$ $\vec{c} = \hat{i} + t \hat{k}$ Let $V(t)$ be the volume of the tetrahedron formed by the origin and the points with position vectors $\vec{a}, \vec{b}, \vec{c}$.

Find the value of $\lim_{t \to 1^-} \frac{dV}{dP}$.

Let $S$ be the set of all $3 \times 3$ matrices of the form $B = \begin{pmatrix} a & b & 0 \\ c & d & 0 \\ 0 & 0 & 1 \end{pmatrix}$, where $a, b, c, d \in \{1, 2, 3, 4, 5, 6\}$. A matrix $B$ is chosen at random from $S$. Let $P$ be the probability that $B$ is non-singular.

Consider the matrix $A(x) = \begin{pmatrix} 0 & x^2 - x & \sin(\pi x) \\ x - x^2 & 0 & e^{x-1} - 1 \\ -\sin(\pi x) & 1 - e^{x-1} & 0 \end{pmatrix}$ and let $f(x) = \det(B A(x) B^T)$.

Find the value of $648 P + f'(2024)$.

Let $X$ and $Y$ be independent continuous random variables, each having the probability density function $f(x)$ defined on the interval $[0, 2]$ by: $f(x) = \frac{d^2}{dx^2} \left( \int_{0}^{x} \frac{(x - t)^2}{2} |t - 1| dt \right)$ Let $E_1$ be the event that the quadratic equation $z^2 - 2Xz + X = 0$ has real roots, and $E_2$ be the event that the quadratic equation $z^2 - 2Yz + Y = 0$ has real roots. Find the probability of the event $E_1 \cup E_2$.

Let $L$ be the horizontal asymptote of the curve $y = \frac{3000t^3 + 2t^2 - 7}{t^3 - t + 4}$ as $t \to \infty$. Two boxes contain mathematical expressions. Box I contains the expression $\left(x^2 + \frac{1}{x}\right)^{15}$ and Box II contains the expression $\left(x^3 - \frac{2}{x^2}\right)^{10}$. A fair coin is tossed. If it shows heads, Box I is selected; if tails, Box II is selected. Then, an integer $r$ is chosen at random from the set $S = \{0, 1, 2, \dots, 30\}$. What is the probability that the coefficient of $x^r$ in the expression of the selected box is strictly greater than $L$?

Let $A, B,$ and $C$ be three events in a sample space such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$, $P(C) = \frac{1}{4}$, $P(A \cap B) = \frac{1}{4}$, $P(B \cap C) = \frac{1}{6}$, $P(C \cap A) = \frac{1}{8}$, and $P(A \cap B \cap C) = \frac{1}{12}$. Let $M$ be a $3 \times 3$ matrix such that its determinant is given by $|M| = P(\overline{A} \cap \overline{B} \cap \overline{C})$. If $N = \text{adj}(2M)$, find the value of $584 \sqrt{|N|}$.

A real number $x$ is chosen uniformly at random from the interval $[-1, 5]$. Let $f(x) = |x^2 - 4x + 3|$. Let $E_1$ be the event that $f(x) \ge 3$ and $E_2$ be the event that $f(x) \le \frac{24}{25}$. Find the probability of the event $E_1' \cap E_2'$.

Let $S$ be a set of 10 points in 3D space consisting of the 4 vertices and the 6 edge midpoints of a regular tetrahedron. Four points are chosen at random from $S$. Let $A$ be the event that the 4 chosen points are coplanar, and $B$ be the event that no three of the chosen points are collinear. Find the probability of the event $A^c \cup B^c$.

Let $X$ be a continuous random variable with the probability density function: $f(x) = \begin{cases} \frac{1}{2} \left(1 + \sin\left(\frac{\pi x}{2}\right)\right), & x \in [-1, 1] \\ 0, & \text{otherwise} \end{cases}$ Let $A$ and $B$ be events defined by $A = \{X > 0\}$ and $B = \{X < -1/3\}$. Calculate the probability $P(A^c \cap B^c)$.

Let $X$ be a continuous random variable with probability density function $f(x) = \begin{cases} a x^2 + b x + c & \text{for } x \in [-1, 1] \\ 0 & \text{otherwise} \end{cases}$, where $a, b, c$ are real constants. What is the maximum possible value of the expected value $E[X]$?

Let $\Omega = \{1, 2, 3, 4, 5, 6, 7\}$ be a sample space equipped with a valid probability measure $P$. Consider the events $E_1 = \{1, 2, 3, 4\}$, $E_2 = \{3, 4, 5, 6\}$, and $E_3 = \{1, 2, 5, 6\}$. It is given that $E_1, E_2$, and $E_3$ are pairwise independent events and $P(E_1) = P(E_2) = P(E_3) = p > 0$. What is the value of $P(\{7\})$?

Let $P = \begin{bmatrix} p_1 \\ p_2 \\ p_3 \end{bmatrix}$ be a probability vector such that $p_i \ge 0$ for $i=1,2,3$ and $\sum_{i=1}^3 p_i = 1$. Consider the skew-symmetric matrix $A = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{bmatrix}$ and the column vector $B = \begin{bmatrix} -1 \\ 2 \\ -1 \end{bmatrix}$. If $P$ satisfies the system of linear equations $AP = B$, then the value of $81(p_1^2 + p_2^2 + p_3^2)$ is:

Three independent trials are performed. In each trial, a fair die is rolled. If the die shows a multiple of $3$, a success is recorded. Let $x, y, z \in \{1, -1\}$ such that the variable takes the value $1$ if the corresponding trial (1st, 2nd, 3rd) is a success, and $-1$ if it is a failure. Consider the vectors $\vec{u} = \hat{i} + \hat{j}$, $\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}$, and $\vec{w} = \hat{i} + \hat{k}$. Let $A$ be the event that the volume of the parallelepiped formed by $\vec{u}, \vec{v}, \vec{w}$ is exactly $1$. Given that event $A$ has occurred, what is the probability that the acute angle between the planes containing the pairs of vectors $(\vec{u}, \vec{v})$ and $(\vec{v}, \vec{w})$ is $\frac{\pi}{6}$?

Four married couples (8 distinct people) are to be seated around a circular table. Let $X$ be the number of arrangements in 8 distinct, numbered chairs such that no couple sits diametrically opposite to each other. Let $Y$ be the number of arrangements in 8 identical, unnumbered chairs (where arrangements are identical if they differ only by rotation) such that exactly one couple sits diametrically opposite to each other. Find the value of $\frac{X}{Y}$.

An experiment has a countably infinite sample space $S = \{E_1, E_2, E_3, \dots\}$. The probability of each elementary event $E_n$ is given by $P(E_n) = \frac{\alpha}{2^n} + \frac{\beta \cdot 3^n}{(3^n - 1)(3^{n+1} - 1)}$ where $\alpha$ and $\beta$ are positive constants. If $\lim_{n \to \infty} 2^n P(E_n) = \frac{1}{4}$, then the probability of the event $E_2$ is:

A real number $x$ is chosen uniformly at random from the interval $[0, 3]$. Let $E_1$ be the event that $[x^2] - 3[x] + 2 \le 0$, and $E_2$ be the event that $x^2 - 3x + 2 \le 0$, where $[\cdot]$ denotes the greatest integer function. The probability of the event $E_1 \cup E_2$ is:

Let $S$ be the sample space of all positive integer solutions $(x, y, z)$ to the equation $x + y + z = 50$. A solution $(x, y, z)$ is chosen at random from $S$. Let $E_1$ be the event that $x, y, z$ can form the side lengths of a triangle (including degenerate triangles where the vertices are collinear). Let $E_2$ be the event that $x, y, z$ can form the side lengths of a non-degenerate triangle (with positive area). Find the conditional probability $P(E_2 \mid E_1)$.