The probability distribution of a random variable $x$ is, $P(x) =\frac{k}{2^x},x= 0,1,2,3$. Then

Match List-I with List-II

Choose the correct answer from the options given below.

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

The correct answer is Option (4) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)

$P(x)=\frac{k}{2^x},\;x=0,1,2,3$.

Since total probability is $1$:

$\displaystyle \frac{k}{2^0}+\frac{k}{2^1}+\frac{k}{2^2}+\frac{k}{2^3} = k\left(1+\frac12+\frac14+\frac18\right)=k\cdot\frac{15}{8}=1$

$\displaystyle k=\frac{8}{15}$

Now:

$P(x=1)=\frac{k}{2}=\frac{8}{15}\cdot\frac12=\frac{4}{15}$

$P(1

$P(x\ge 2)=P(2)+P(3)=\frac{k}{4}+\frac{k}{8} =\frac{2}{15}+\frac{1}{15}=\frac{3}{15}=\frac15$

Correct matching: A–III, B–IV, C–I, D–II