Question

Let \(X\) denote the number of hours a person drive during a randomly selected working day. The probability that \(X\) can take the value \(x_i\) has the following form, where \(k\) is some unknown constant.
\[ P(X = x_i) = \begin{cases} 0.2, & \text{if } x_i = 0 \\ kx_i, & \text{if } x_i = 1 \text{ or } 2 \\ k(4 - x_i), & \text{if } x_i = 3 \\ 0, & \text{otherwise} \end{cases} \]
Match the LIST-I with LIST-II

LIST-I	LIST-II
A. The value of \(k\)	I. 1
B. The probability that the person drive atleast two hours on a selected working day	II. 0.4
C. The probability that the person drive atmost three hours on a selected working day	III. 0.2
D. The probability that the person drive atmost one hour on a selected working day	IV. 0.6

Choose the correct answer from the options given below:

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

Hint:

Total probability is 1, so 0.2 + 4k = 1 and k = 0.2. Then add the required probabilities.

Answer 1

The Correct Option is B

Step 1: Tabulate the distribution.
Write the four values of X with their probabilities: 0 gives 0.2, 1 gives $k$, 2 gives $2k$ and 3 gives $k(4-3)=k$. Any other value has probability 0.

Step 2: Use the total rule.
For a valid distribution the sum of all probabilities is 1, so $0.2 + 4k = 1$. This gives $4k = 0.8$ and $k = 0.2$. So the value of $k$ is 0.2, which is entry III. A goes with III.

Step 3: Use the complement for B.
Instead of adding, subtract the small values from 1. $P(X \geq 2) = 1 - P(X \leq 1) = 1 - (0.2 + 0.2) = 1 - 0.4 = 0.6$. So B goes with IV.

Step 4: Value for D.
From the same calculation, $P(X \leq 1) = 0.2 + 0.2 = 0.4$. So D goes with II.

Step 5: Value for C.
The largest value X can take is 3, so $P(X \leq 3)$ covers the whole sample space and equals 1. So C goes with I.

Step 6: Read off the option.
The pairs are A-III, B-IV, C-I, D-II. Only option 2 has exactly these pairs.

Final Answer:
The correct matching is A-III, B-IV, C-I, D-II. \[\boxed{\text{Option (2)}}\]