Question

$\text{Let } X \text{ denote the number of hours you play during a randomly selected day. The probability that } X \text{ can take values } x \text{ has the following form, where } c \text{ is some constant:}$
$P(X = x) = \begin{cases} 0.1, & \text{if } x = 0 \\\ cx, & \text{if } x = 1 \text{ or } x = 2 \\\ c(5 - x), & \text{if } x = 3 \text{ or } x = 4 \\\ 0, & \text{otherwise} \end{cases}$
$\text{Match List-I with List-II:}$

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

Answer 1

Answer: (B)

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

Answer 2

The sum of all probabilities must equal 1:

$P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 1.$

Substitute the given probabilities:

$0.1 + c(1) + c(2) + c(2) + c(1) = 1.$

Simplify:

$0.1 + 6c = 1 \Rightarrow 6c = 0.9 \Rightarrow c = 0.15.$

(A) $c = 0.15$. Match: (A) -> (IV).

(B) $P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)$:

$P(X \leq 2) = 0.1 + c(1) + c(2) = 0.1 + 0.15 + 0.3 = 0.55.$

Match: (B) -> (III).

(C) $P(X = 2) = c(2) = 0.3.$ Match: (C) -> (II).

(D) $P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4)$:

$P(X \geq 2) = c(2) + c(2) + c(1) = 0.3 + 0.3 + 0.15 = 0.75.$

Match: (D) -> (I).

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