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 at least two hours on a selected working day | II. | 0.4 |
| C. | The probability that the person drive at most three hours on a selected working day | III. | 0.2 |
| D. | The probability that the person drive at most one hour on a selected working day | IV. | 0.6 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → A-III, B-IV, C-I, D-II
From the given form, the distribution is
\( P(X=0)=0.2,\ P(X=1)=k \)
\( P(X=2)=2k,\ P(X=3)=k(4-3)=k \)
(A) The value of k
This matches (III) 0.2. The probabilities must add to 1:
\( 0.2+k+2k+k=1 \)
\( 4k=0.8 \)
\( k=0.2 \)
So the distribution is P(0) = 0.2, P(1) = 0.2, P(2) = 0.4, P(3) = 0.2.
(B) at least two hours
This matches (IV) 0.6.
\( P(X\ge 2)=P(2)+P(3)=0.4+0.2=0.6 \)
(C) at most three hours
This matches (I) 1. X never exceeds 3, so this is the sure event:
\( P(X\le 3)=0.2+0.2+0.4+0.2=1 \)
(D) at most one hour
This matches (II) 0.4.
\( P(X\le 1)=P(0)+P(1)=0.2+0.2=0.4 \)
So the matching is A-III, B-IV, C-I, D-II.