If the random variable X has the following distribution:, X, 0, 1, 2, otherwise, P(X), k, 2k, 3k, 0, Match List-I with List-II:, List-I, List-II, (A) k, (I) 5/6, (B) P(X < 2), (II) 4/3, (C) E(X), (III) 1/2, (D) P(1 ≤ X ≤ 2), (IV) 1/6 Option 1: (A) - (I), (B) - (II), (C) - (III), (D) - (IV) Option 2: (A) - (IV), (B) - (III), (C) - (II), (D) - (I) Option 3: (A) - (I), (B) - (II), (C) - (IV), (D) - (III) Option 4: (A) - (III), (B) - (IV), (C) - (I), (D) - (II)

Answer: (A) - (IV), (B) - (III), (C) - (II), (D) - (I). Solution: First, we need to find the value of k. For any probability distribution, the sum of all probabilities must equal 1: P(X = 0) + P(X = 1) + P(X = 2) = 1 k + 2k + 3k = 1 6k = 1 k = 1/6 Next, let's find P(X For the expected value E(X): E(X) = [x * P(X = x)] E(X) = 0 * k + 1 * 2k + 2 * 3k E(X) = 2k + 6k = 8k = 8 * 1/6 = 4/3 For P(1 ≤ X ≤ 2): P(1 ≤ X ≤ 2) = P(X = 1) + P(X = 2) P(1 ≤ X ≤ 2) = 2k + 3k = 5k = 5 * 1/6 = 5/6 Matching the lists: (A) k = 1/6 matches with (IV) (B) P(X < 2) = 1/2 matches with (III) (C) E(X) = 4/3 matches with (II) (D) P(1 ≤ X ≤ 2) = 5/6 matches with (I)

Algebra

Probability

Easy

core

If the random variable $X$ has the following distribution:

$X$ 0 1 2 otherwise
$P(X)$ $k$ $2k$ $3k$ $0$

Match List-I with List-II:

List-I List-II
(A) $k$ (I) $\frac{5}{6}$
(B) $P(X < 2)$ (II) $\frac{4}{3}$
(C) $E(X)$ (III) $\frac{1}{2}$
(D) $P(1 \leq X \leq 2)$ (IV) $\frac{1}{6}$

$X$	0	1	2	otherwise
$P(X)$	$k$	$2k$	$3k$	$0$

List-I	List-II
(A) $k$	(I) $\frac{5}{6}$
(B) $P(X < 2)$	(II) $\frac{4}{3}$
(C) $E(X)$	(III) $\frac{1}{2}$
(D) $P(1 \leq X \leq 2)$	(IV) $\frac{1}{6}$

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

(A) - (I), (B) - (II), (C) - (IV), (D) - (III)

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)

✅ Correct Option: 2

First, we need to find the value of $k$ .

For any probability distribution, the sum of all probabilities must equal 1:

$P(X = 0) + P(X = 1) + P(X = 2) = 1$

$k + 2k + 3k = 1$

$6k = 1$

$k = \dfrac{1}{6}$

Next, let's find $P(X < 2)$ :

$P(X < 2) = P(X = 0) + P(X = 1)$

$P(X < 2) = k + 2k = 3k = 3 \cdot \dfrac{1}{6} = \dfrac{1}{2}$

For the expected value $E(X)$ :

$E(X) = \sum[x \cdot P(X = x)]$

$E(X) = 0 \cdot k + 1 \cdot 2k + 2 \cdot 3k$

$E(X) = 2k + 6k = 8k = 8 \cdot \dfrac{1}{6} = \dfrac{4}{3}$

For $P(1 \leq X \leq 2)$ :

$P(1 \leq X \leq 2) = P(X = 1) + P(X = 2)$

$P(1 \leq X \leq 2) = 2k + 3k = 5k = 5 \cdot \dfrac{1}{6} = \dfrac{5}{6}$

Matching the lists:

(A) $k = \dfrac{1}{6}$ matches with (IV)

(B) $P(X < 2) = \dfrac{1}{2}$ matches with (III)

(D) $P(1 \leq X \leq 2) = \dfrac{5}{6}$ matches with (I)

2025: 19 May Shift 1

If X is a normal variate with mean 16 and standard deviation 4, then the value of standard normal variate Z corresponding to X = 17 is:

2026: 14 May Shift 2

A pair of coins thrown then

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	Probability of occurance of head on both coins is	I.	$\dfrac{3}{4}$
B.	Probability of occurance of at least one tail is	II.	$\dfrac{1}{2}$
C.	Probability of occurance of one tail only is	III.	$1$
D.	Probability of a tail appears on one coin given that only one coin shows head.	IV.	$\dfrac{1}{4}$

Choose the correct answer from the options given below:

2025: 16 May Shift 1

The random variable $x$ has a probability distribution $p(x)$ of the form $P(x=r)=\begin{cases} rk, & \text{if } r \le 2, \\ (r-1)k, & \text{if } 2<r\le 4, \\ 0, & \text{otherwise,} \end{cases}$ where $r \in \mathbb{N}\cup\{0\}$ and $k\in\mathbb{R}$ , where $\mathbb{N}$ is the set of natural numbers. Then:

(A) $k=\frac{1}{9}$

(B) $P(2\le x\le 3)=\frac{1}{2}$

(D) $P(x>1)=\frac{7}{8}$

Choose the correct answer from the options given below.

2025: 22 May Shift 1

A random variable X has the following probability distribution:

X	-2	-1	0	1	2	3
P(X)	0.1	0.2	k	0.3	2k	0.1

then which of the following are TRUE?

(A) $k=0.1$

(B) $P(X < 1) = 0.4$

(D) $P(0 < X < 3) = 0.5$

Choose the correct answer from the options given below:

2026: 18 May Shift 2

One ticket is drawn from a purse containing four tickets with numbers 223, 133, 212, 322. Let $A_i$ be the event that i $^{th}$ digit of the number on drawn ticket is 2. Then which of the following events are independent ?

A. $A_1$ and $A_2$

B. $A_2$ and $A_3$

C. $A_1$ and $A_3$

D. $A_1$ , $A_2$ and $A_3$

Choose the correct answer from the options given below:

2026: 22 May Shift 1

A school has to send the report cards of 3 students, but the cleark has not paid attention to match the address on the envelope with the student report card is: The probability that exactly one of the students received his or her own report card is:

Solution

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