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

The Correct Option is B

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).