Probability - Class 12

Two fair dice are rolled to obtain two numbers $a$ and $b$ respectively, where $a, b \in \{1, 2, 3, 4, 5, 6\}$. Consider the system of linear equations: $x + 2y + z = 1$ $2x + ay + 3z = b$ $3x + 7y + 4z = 4$ Three urns $U_1, U_2,$ and $U_3$ contain balls as follows: $U_1$ has 3 Red and 2 Black balls, $U_2$ has 2 Red and 3 Black balls, and $U_3$ has 4 Red and 1 Black balls. A ball is drawn at random from $U_1$ if the system has a unique solution, from $U_2$ if the system has infinitely many solutions, and from $U_3$ if the system has no solution. If the drawn ball is Red, what is the probability that the system of equations had no solution?

A biased die has its faces painted with three colors: Red, Blue, and Green. The probability of rolling a Red face is $\frac{1}{2}$, a Blue face is $\frac{1}{3}$, and a Green face is $\frac{1}{6}$. In a game, a player rolls this die $6$ times independently. For each Red face rolled, the player scores $1$ point; for each Blue face, $2$ points; and for each Green face, $0$ points. What is the probability that the player scores exactly $8$ points in the $6$ rolls?

Let $G(x)$ be a generating function whose first derivative is given by $G'(x) = x^4 - 12x^3 + 46x^2 - 60x$. A random variable $X$ follows a binomial distribution $B(n, p)$ with $n=9$ and $p=\frac{1}{2}$. Let $A$ be the event that $G(x)$ is strictly decreasing at $x = X$, and let $B$ be the event that $G(x)$ is strictly concave at $x = X$. Find the conditional probability $P(B \mid A)$.

Let $S = \{1, \omega, \omega^2\}$, where $\omega = e^{i2\pi/3}$. Let $\mathcal{F}$ be the set of all quadratic equations of the form $z^2 + az + b = 0$, where $a, b \in S$. If two distinct equations are chosen at random from $\mathcal{F}$, the probability that they have at least one common root is $\frac{p}{q}$, where $p$ and $q$ are coprime positive integers. The value of $p+q$ is:

Let $S = \{-2, -1, 0, 1, 2\}$. A point $Q(a, b, c)$ is chosen uniformly at random such that $a, b, c \in S$. Let $P$ be the point $(1, 1, 1)$ and $\Pi$ be the plane $x + y + z = 0$. What is the probability that the line segment $PQ$ intersects the plane $\Pi$?

15 identical coins are distributed into 3 distinct boxes. Let $x_i$ be the number of coins in the $i$-th box ($i=1,2,3$). Let $E$ be the event that $\sum_{i=1}^3 \left[ \frac{x_i}{4} \right] = 2$, where $[ \cdot ]$ denotes the greatest integer function. Let $F$ be the event that $x_i \in [0, 7)$ for all $i=1,2,3$. The conditional probability $P(F|E)$ is

Let $I = \int_0^{\pi} x \sin^6 x \cos^4 x \, dx$. A continuous random variable $X$ has the probability density function $f(x) = \begin{cases} \frac{512}{3\pi^2 \ln 3} \cdot \frac{I}{x^2+2x} & \text{for } x \ge \lambda \\ 0 & \text{otherwise} \end{cases}$ where $\lambda > 0$ is a constant such that $f(x)$ is a valid probability density function. Find the conditional probability $P\left(X > 2 \mid X > \frac{1}{2}\right)$.

Let $S$ be the set of distinct complex roots of the polynomial $P(z) = z^6 - z^5 + z^4 - z^3 + z^2 - z + 1$. For any $z \in S$, let $\arg z \in (-\pi, \pi]$ denote its principal argument. Define the set $T \subset S$ as: $T = \left\{ z \in S \ \Big| \ \sum_{n=1}^{\infty} \left( \frac{\arcsin(\sin(\arg z))}{\arg z} \right)^n \text{ diverges} \right\}$ A pair of distinct complex numbers $\{z, w\}$ is chosen uniformly at random from $T$. Let $p$ be the probability that $\text{Im}(z + w) > 0$. Find the value of: $p \sum_{z \in S} |z^2 - z + 1|^2$

Let $\hat{a}, \hat{b}, \hat{c}$ be three mutually perpendicular unit vectors in $\mathbb{R}^3$ forming a right-handed coordinate system. Let $\hat{x}$ be a random unit vector chosen uniformly from the surface of the unit sphere centered at the origin. Let $V$ be the volume of the parallelepiped whose coterminous edges are the vectors $\hat{a}-\hat{x}, \hat{b}-\hat{x}$, and $\hat{c}-\hat{x}$.

Define two events based on this volume: $E_1$: $V \le 2$ $E_2$: $V \ge 1$

Find the conditional probability $P(E_2 \mid E_1)$.

Let $p = \lim_{n \to \infty} \sum_{k=1}^n \frac{k^2}{n^3 + k^2 \sin(k)}$. A biased coin with probability of heads $P(H) = \frac{1}{4}$ is tossed repeatedly. Let $N$ be the random variable representing the number of tails obtained before exactly 3 heads appear. Let $X$ be the number of successes in $N$ independent Bernoulli trials, where the probability of success in each trial is $p$. Find the variance of the random variable $Z = X + \frac{N}{3}$.

Let $S$ be the sample space of all $2 \times 2$ orthogonal matrices with real entries. A matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ is chosen uniformly at random from $S$. (The uniform distribution is defined by choosing the parameter $\theta \in [0, 2\pi)$ uniformly, and the determinant $\det(A) \in \{-1, 1\}$ with equal probability $\frac{1}{2}$). Let $E_1$ be the event that $|a_{11} + a_{22}| > \sqrt{2}$, $E_2$ be the event that $|a_{12} - a_{21}| > \sqrt{2}$, $E_3$ be the event that $|a_{11}| \ge \frac{1}{2}$, and $E_4$ be the event that $\det(A) = 1$. Which of the following statements is correct?

Let $A$ and $B$ be two independent events with probabilities $P(A) = \frac{1}{2}$ and $P(B) = q$, where $q \in (0, 1)$. Consider the matrix $M = \begin{pmatrix} P(A \cup B) & P(A^c \cap B^c) & -\frac{1}{2} \\ P(A \cap B) & P(A^c \cup B^c) & -\frac{1}{2} \\ P(A) & P(A^c) & -\frac{1}{2} \end{pmatrix}$ Let $S$ be the set of all values of $q$ for which the eigenvectors of $M$ span $\mathbb{R}^3$. Then the sum of all elements in $S$ is:

Let $A$ and $B$ be two disjoint events in a sample space such that $P(A) = 3P(B)$. Consider three points in the Cartesian plane: $V_1 = (P(A), P(B))$, $V_2 = (P(A \cup B), P(A \cap B))$, and $V_3 = (P(A^c), P(B^c))$. Let $M$ be the $3 \times 3$ matrix whose rows are $(x_i, y_i, 1)$ for $i=1,2,3$, where $(x_i, y_i)$ are the coordinates of $V_i$ in the standard $(X, Y)$ coordinate system. Let $D = \det(M)$. A new coordinate system $(X', Y')$ is defined by the transformation $X' = X - P(A)$ and $Y' = P(B) - Y$. Let $M'$ be the corresponding matrix using the coordinates of the same points in the $(X', Y')$ system, and let $D' = \det(M')$. If $D - D' = \frac{1}{4}$, what is the value of $P(A \cup B)$?

For sufficiently small $x > 0$, let $A_x$ and $B_x$ be two events such that $P(A_x) = x$ and $P(B_x) = x^2$. The probability of their union satisfies the integral equation: $\int_0^x t P(A_t \cup B_t) dt = \frac{x^3}{3} + \frac{x^4}{4} - \frac{153}{5} x^5 - \int_0^x \frac{x^2 - t^2}{2} t^2 \sin\left(\frac{1}{t}\right) dt$ Let $L = \lim_{x \to 0^+} \frac{P(A_x \cap B_x)}{P(A_x) P(B_x)}$. Find the value of $L$.

Let $A = \{1, 2, 3, 4\}$ and $S$ be the set of all functions from $A$ to $A$. The set $S$ is partitioned into three subsets $S_1, S_2, S_3$ as follows:

$S_1$: The set of all bijective functions. $S_2$: The set of all functions $h \in S$ that can be expressed as $h = f \circ g$, where $g: A \to B$ is surjective, $f: B \to A$ is injective, and $B = \{1, 2, 3\}$. $S_3$: The set of all remaining functions in $S$.

A biased die is rolled to select one of the subsets, such that the probability of selecting $S_i$ is proportional to $i$. Then, a function $h$ is chosen uniformly at random from the selected subset. Given that the chosen function satisfies $h(1) = h(2)$, what is the probability that it was chosen from $S_2$?

Let $I_n$ be the set of all integers $k$ ($0 \le k \le n$) for which the binomial coefficient $\binom{n}{k}$ attains its maximum value. Define $S_n = \sum_{k \in I_n} k$. Two bags $B_1$ and $B_2$ are given. Bag $B_1$ contains 2 Red and 3 Black balls, while Bag $B_2$ contains 3 Red and 4 Black balls. A bag is selected at random such that the probability of choosing $B_1$ is proportional to $\sum_{m=1}^{7} S_{2m}$ and the probability of choosing $B_2$ is proportional to $\sum_{m=1}^{8} S_{2m-1}$. A ball is drawn at random from the selected bag and is found to be Red. The probability that it came from Bag $B_1$ is:

A point $P(x,y,z)$ is chosen uniformly at random from the solid sphere $x^2 + y^2 + z^2 \le 9$. Let $D$ be the perpendicular distance from $P$ to the plane $\Pi: x + 2y + 2z = 3$. Two boxes, $B_1$ and $B_2$, contain colored balls. Box $B_1$ contains 2 white and 3 black balls. Box $B_2$ contains 4 white and 1 black balls. If $D < 1$, Box $B_1$ is selected. If $D \ge 1$, Box $B_2$ is selected. A ball is drawn at random from the selected box and is found to be white. The probability that Box $B_1$ was selected is:

A point $z$ is chosen uniformly at random from the region $S = \{z \in \mathbb{C} : 4 \le |z| \le 5\}$. Let $w = u + iv$ be a complex number defined by the transformation $w = z + \frac{16}{z}$. A Bernoulli trial is considered a success if $w$ satisfies both of the following conditions: (i) $\det \begin{pmatrix} u & v & 1 \\ 8 & 0 & 1 \\ -8 & 0 & 1 \end{pmatrix} \ge 0$ (ii) $|w - 8| + |w + 8| \le 12$ Let $X$ be the number of successes in $n = 2024$ independent trials. Find the value of $E(X^2 + X + 1)$, where $E(\cdot)$ denotes the expected value.

Let $p$ be a valid probability of success for a Bernoulli trial such that the function $f(x) = 16x^4 - 48x^3 + 46x^2 - 15x$ attains a local maximum at $x=p$. An experiment consists of repeatedly performing independent Bernoulli trials with success probability $p$ until 2 consecutive successes occur. Let $E$ be the expected number of trials required. Find the value of $9E$.

Let $X$ be a random variable representing the number of successes in $25$ independent Bernoulli trials, each with probability of success $p \in (0, 1)$. If the coefficient of variation of $X$ is at most $\frac{\sqrt{2}}{10}$, then the maximum possible standard deviation of $X$ is:

Let $V(x)$ be the volume of a right circular cone of height $x$ inscribed in a sphere of radius $3$. Let $x_0$ be the height that maximizes this volume, and let $\lambda = \frac{9}{32\pi} V(x_0)$. For each integer $n \ge 3$, let $X_n$ be a random variable following the binomial distribution $B(n, p_n)$, where $p_n = \frac{\lambda}{n}$. Evaluate the limit: $\lim_{n \to \infty} n \left( e^\lambda P(X_n \le 1) - 4 \right)$

Let $A = \begin{pmatrix} 3p - 2 & \tan \theta \\ -\cot \theta & q \end{pmatrix}$, where $p \in (0, 1)$, $q \in \mathbb{R}$, and $\theta \in \left(0, \frac{\pi}{2}\right)$. It is given that $A$ is a nilpotent matrix of index 2. If $p$ represents the probability of success in a single trial of a binomial experiment, find the probability of at least 4 successes in 6 independent trials.

Let $S$ be the feasible region in the first quadrant ($x \ge 0, y \ge 0$) satisfying the system of inequalities: $-x - 2y \ge -6$ $2x + y \le 6$ $3x + 4y \le 24$ Let $M$ be the maximum value of the objective function $Z = x + 2y$ over the region $S$. A biased coin, whose probability of showing heads is $p = \frac{M}{15}$, is tossed 5 times. Let $X$ be the number of heads obtained. Find the probability that the coin shows heads at least 3 times.

Let $X$ be a random variable representing the number of successes in $n = 280$ independent Bernoulli trials. The probability of success in a single trial is given by $p$, where $p = \int_0^{\pi/2} \frac{2\sin^{280} x + \sin(2x)}{2(\sin^{280} x + \cos^{280} x + \sin(2x))} \, dx$ Find the value of $\frac{16}{\pi^2} E[X^2 - X]$.

Let $V = \{ \vec{u}, \vec{v}, \vec{w} \}$ be a set of three vectors where $\vec{u} = [1, -2, 1]$, $\vec{v} = [-1, 1, 2]$, and $\vec{w} = [2, -1, -3]$. Let $S$ be the set of all $3 \times 3$ matrices whose rows are formed by choosing vectors from $V$ with replacement. A matrix $A$ is chosen at random from $S$. Let $E_1$ be the event that the system of linear equations $A \vec{x} = \vec{0}$ has non-trivial solutions, and $E_2$ be the event that $A$ is a nilpotent matrix. The value of $P(E_2 \mid E_1)$ is:

Let $S = \{1, 2, 3, \dots, 1500\}$. A number $n$ is chosen uniformly at random from $S$. Let $E$ be the event that $\sum_{k=1}^{n} (302 - |k - 302|) \ge 0$ Find the probability of the event $E$.

Let $S = \{1, 2, 3, \dots, 15\}$. A number $k$ is chosen at random from $S$. Let the function $f_k : [-3, 3] \to \mathbb{R}$ be defined as $f_k(x) = \min(9-x^2, 2|x|+k)$ Let $M_k$ be the global maximum value of $f_k(x)$ on the interval $[-3, 3]$. Let $E$ be the event that $M_k$ is an integer, and $F$ be the event that $f_k(x)$ is non-differentiable at the point(s) where it attains its global maximum. Find the conditional probability $P(F \mid E)$.

Let $S = \{(h, k) \in \mathbb{Z} \times \mathbb{Z} : h^2 + k^2 \le 34\}$. A point $(h, k)$ is chosen at random from $S$. Given that the equation $hx^2 - hy^2 + 2kxy + 2x - 2y + 1 = 0$ represents a conic section, what is the probability that it represents a non-degenerate rectangular hyperbola?

Let $S = \{-2, -1, 0, 1, 2\}$. An ordered triplet $(x, y, z)$ of distinct numbers is chosen randomly from $S$ to form the matrix $A = \begin{pmatrix} x & |x| & 1 \\ y & |y| & 1 \\ z & |z| & 1 \end{pmatrix}$ Let $E$ be the event that $\det(\text{adj} A) \ge 16$ and $F$ be the event that the matrix $A$ is invertible. The value of the conditional probability $P(E \mid F)$ is

Let $S$ be the sample space of all functions $f: A \to A$, where $A = \{1, 2, 3, 4\}$. A function is chosen at random from $S$ with uniform probability. Define an equivalence relation $\sim$ on $S$ by $f \sim g \iff \text{Range}(f) = \text{Range}(g)$. Let $C_f$ denote the equivalence class containing $f$. Let $E$ be the event that the chosen function satisfies $f(f(x)) = f(x)$ for all $x \in A$. Let $F$ be the event that the chosen function belongs to an equivalence class of size exactly $16$. Find the conditional probability $P(E \mid F)$.

Let $X$ be a continuous random variable taking values in the interval $[-1, 1]$. Its probability density function $f(x)$ satisfies the integral equation: $\int_{0}^{x} f(t) \sqrt{1+t^2} \, dt = \frac{1}{4} \left( x \sqrt{1+x^2} + \ln(x + \sqrt{1+x^2}) \right) + \frac{1}{6} \left( (1+x^2)^{3/2} - 1 \right)$ for all $x \in [-1, 1]$. Let $E_1$ be the event that the roots of the quadratic equation $y^2 + 2Xy + X = 0$ (in variable $y$) are real. Let $E_2$ be the event that $X$ satisfies the inequality: $\int_{-X}^{X} \frac{t^3 \cos^2 t}{1+t^2} \, dt + X^2 \le \frac{1}{4}$ Find the conditional probability $P(E_1 \mid E_2)$.

A line $L$ with direction vector $\vec{d}$ is generated such that $\vec{d}$ is a random vector uniformly distributed on the surface of the unit sphere in $\mathbb{R}^3$. Let $P$ be a fixed plane in $\mathbb{R}^3$. Let $\theta \in \left[0, \frac{\pi}{2}\right]$ be the acute angle between the line $L$ and the plane $P$. If $E_1$ is the event that $\theta \ge \frac{\pi}{6}$ and $E_2$ is the event that $\theta \le \frac{\pi}{3}$, then the conditional probability $P(E_2 \mid E_1)$ is equal to

Let $X$ be a discrete random variable taking values in the set of non-negative integers $\mathbb{W} = \{0, 1, 2, \dots\}$. The probability mass function of $X$ is given by $P(X=k) = C \binom{k+2}{2} t^k$ where $t = \frac{1}{\pi} \sin^{-1}\left(\sin \frac{8\pi}{9}\right)$ and $C$ is a positive constant such that $\sum_{k=0}^{\infty} P(X=k) = 1$. Find the conditional probability $P(X=1 \mid X \le 2)$.

Let $A$ and $B$ be two independent events associated with a random experiment. The probabilities of these events are given by: $P(A) = \frac{1}{3^{20}} \sum_{r=0}^{10} \binom{20}{2r} 2^{2r}$ $P(B) = \frac{1}{2^{20}} \sum_{r=0}^{20} \binom{20}{r} \sin^2\left(\frac{r\pi}{2}\right)$ Find the value of $\lfloor 4 \cdot P(A \cup B) \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.

The probability $P(t)$ that a quantum system remains in its initial state at time $t \ge 0$ satisfies the differential equation $\frac{dP}{dt} = -\frac{P^2}{2tP + 1}$ with the initial condition $P(0) = 1$. A researcher measures the state at $t=2$ and correctly determines the probability $P(2)$. To predict the probability at $t=12$, the researcher incorrectly assumes the system's decay is memoryless, satisfying $P(t_1 + t_2) = P(t_1)P(t_2)$ for all $t_1, t_2 \ge 0$. Let $P_{est}$ be the researcher's estimated probability at $t=12$, and $P_{true}$ be the actual probability at $t=12$ governed by the differential equation. Find the value of $\frac{P_{true}}{P_{est}}$.

A biased coin with probability of heads $p \in (0, 1)$ is tossed repeatedly until a head appears. Let $X$ be the total number of tosses required. Let $A$ be the event that $X$ is a multiple of 3, and $B$ be the event that $X \ge 4$. If $P(A \cup B) = \frac{25}{36}$, find the probability that $X$ is an even number.

Let $E_1, E_2,$ and $E_3$ be a partition of a sample space. The prior probabilities of these events are given by: $P(E_1) = \frac{[x]}{5}, \quad P(E_2) = \frac{[x^2]}{5}, \quad P(E_3) = \frac{[x^3]}{5}$ where $x > 0$ and $[\cdot]$ denotes the greatest integer function. Let $S$ be the set of all possible values of $x$ for which this partition is valid. If $S = [a, b)$, an observable event $A$ occurs such that the conditional probabilities are $P(A|E_1) = \frac{1}{2}$, $P(A|E_2) = \frac{3}{4}$, and $P(A|E_3) = \frac{1}{4}$. Find the value of $(a^2 + b^3) P(E_2|A)$.

Let $a_r$ be the coefficient of $x^r$ in the polynomial $P(x) = \frac{1}{2} \left( (1+x)^{13} + (1-x)^{13} \right)$. A set of 14 bags, labeled $B_0, B_1, \dots, B_{13}$, is given. A bag is selected such that the probability of selecting bag $B_r$ is proportional to $a_r$. Each bag $B_r$ contains $r$ red balls and $13-r$ black balls. A bag is chosen at random according to the given probabilities, and two balls are drawn from it, one after another, with replacement. Given that both drawn balls are red, what is the probability that they were drawn from bag $B_6$?

Let $E_1$ and $E_2$ be two mutually exclusive and exhaustive events such that the prior probability $P(E_1)$ is given by $P(E_1) = \frac{6}{13} \lim_{n \to \infty} \frac{1}{n^2} \sum_{k=1}^n \sqrt{ 9k^2 - 6kn + n^2 }$ A box $B_1$ contains 4 red and 1 black balls, while box $B_2$ contains 3 red and 5 black balls. A box is selected such that the probability of selecting $B_1$ is $P(E_1)$ and the probability of selecting $B_2$ is $P(E_2)$. A ball is drawn at random from the selected box and is found to be red. The probability that it was drawn from box $B_2$ is:

A particle $P(x,y)$ is placed in the Cartesian plane. A random process selects its domain of uniform distribution: with probability $\frac{2}{5}$, the particle is distributed uniformly over the region $R_1$ enclosed by the locus $|x| + |y| = 2$; with probability $\frac{3}{5}$, it is distributed uniformly over the region $R_2$ enclosed by the locus $\max(|x|, |y|) = 2$. Given that the particle's position satisfies the condition $x^2 + y^2 \le 4$, what is the probability that its domain was $R_1$?

Let $g(x) = \left| \int_{-1}^x (2|t| - 2) \, dt \right|$. Let $m$ be the number of points in $\mathbb{R}$ where $g(x)$ is non-differentiable. A biased coin with probability of heads $p = \frac{m}{5}$ is tossed 75 times. If $X$ is the number of heads obtained, find the value of $E(X) + \text{Var}(X)$.

Let $\vec{u} = \sec^2 \theta \hat{i} + \cot^2 \theta \hat{j} + \csc^2 \theta \hat{k}$, $\vec{v} = \tan^2 \theta \hat{i} + \csc^2 \theta \hat{j} + \sec^2 \theta \hat{k}$, and $\vec{w} = \hat{i} - \hat{j} + (\cot^2 \theta - \tan^2 \theta)\hat{k}$, where $\theta \in \left(0, \frac{\pi}{4}\right)$. Let $p$ be the volume of the parallelepiped whose coterminous edges are $\vec{u}, \vec{v}, \vec{w}$. A biased coin, which shows heads with probability $p$, is tossed 10 times. Let $X$ denote the number of heads obtained. Find the value of $\sum_{k=0}^{10} (k^2 - 3k + 2) P(X=k) + 2024^p$

Let $A_n = [a_{ij}]$ be an $n \times n$ matrix where $a_{ij} = i + j$ for $1 \le i, j \le n$. Let $p = \frac{|adj(A_n)| + 1}{|A_n|^{n-1} + 2}$ be the probability of success in a single trial of a Bernoulli experiment. Let $X$ be the number of successes in $n$ independent trials of this experiment. If $P(X=2) = P(X=3) > 0$, then the variance of $X$ is equal to

Let $S$ be the set of all positive integers $n$ for which there exists a positive integer $m$ satisfying the system of linear inequalities: $3n - 2m \ge 14$ $n + m \le 6$ A biased coin, whose probability of getting heads is $p = \max_{x > 0} \left( \frac{x^2}{x^4 + 4} \right)$, is tossed $n$ times for each $n \in S$. Let $P_n$ be the probability of getting exactly 2 heads in $n$ tosses. Find the value of $4096 \sum_{n \in S} P_n$.

Let $B$ be a $3 \times 3$ matrix such that $B^2 = I$, $B \neq I$, and $B \neq -I$. Let $A = \frac{1}{2}(I - B)$. The probability of success of an event is given by $p = \det(A)$. A trial is repeated $216$ times independently. Let $X$ be the number of successes, and let $\lambda = P(X \ge 1)$. If $P(x) = \det(A - x I)$ and $Q(x) = \det(B - x I)$, find the value of $P(\lambda) + (Q(\lambda))^2$.

A biased tetrahedral die has four faces coloured red, black, white, and white. The probability of rolling the red face is $\frac{1}{4}$, the black face is $\frac{1}{4}$, and a white face is $\frac{1}{2}$. If the die is rolled $5$ times, let $p$ be the probability that either the red face appears at least twice, or the black face appears at least twice (or both). Find the value of $p$.

Let $Y$ be a random variable chosen uniformly from the set $\{-2, -1, 0, 1, 2\}$. A set of 5 distinct objects is divided into $k$ non-empty subsets, where $k = 3 + \text{sgn}(Y)$. All valid partitions of the 5 objects into $k$ subsets are equally likely. Let $X$ be the random variable representing the number of objects in the subset containing a specific object $O_1$. Find the expected value $E[X]$.