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:
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
Choose the correct answer from the options given below:
\[ P(X = x_i) = \begin{cases} 0.2, & \text{if } x_i = 0 \\ kx_i, & \text{if } x_i = 1 \text{ or } 2 \\ k(4 - x_i), & \text{if } x_i = 3 \\ 0, & \text{otherwise} \end{cases} \]
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 |
Show Hint
Total probability is 1, so 0.2 + 4k = 1 and k = 0.2. Then add the required probabilities.
Updated On: Oct 1, 2026
- A-III, B-II, C-I, D-IV
- A-III, B-IV, C-I, D-II
- A-IV, B-II, C-III, D-I
- A-IV, B-III, C-II, D-I
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
A probability distribution lists every possible value of X with its probability. The probabilities of all values must add up to 1. We use this to find the unknown constant k, then we read the other probabilities from the table.
Step 2: List the probabilities.
For \(x_i=0\): \(P=0.2\).
For \(x_i=1\): \(P=k(1)=k\).
For \(x_i=2\): \(P=k(2)=2k\).
For \(x_i=3\): \(P=k(4-3)=k\).
Step 3: Find k (match A).
The total probability is 1: \[ 0.2 + k + 2k + k = 1 \] \[ 4k = 0.8 \] \[ k = 0.2 \] So A matches III (0.2).
Step 4: Match B.
At least two hours means \(X \geq 2\): \[ P(X \geq 2) = P(2) + P(3) = 2k + k = 3k = 0.6 \] So B matches IV (0.6).
Step 5: Match C.
At most three hours means every possible value 0, 1, 2 and 3, because X cannot take any other value. So the probability is the total, which is 1. So C matches I (1).
Step 6: Match D.
At most one hour means X is 0 or 1: \[ P(0) + P(1) = 0.2 + k = 0.2 + 0.2 = 0.4 \] So D matches II (0.4).
Step 7: Choose the option.
We have A-III, B-IV, C-I, D-II. This is option 2. Option 1 has B-II and D-IV, which is swapped. Options 3 and 4 give A-IV, but k is 0.2, not 0.6, so they are wrong.
Final Answer:
The matching is A-III, B-IV, C-I, D-II. \[ \boxed{\text{Option (2)}} \]
A probability distribution lists every possible value of X with its probability. The probabilities of all values must add up to 1. We use this to find the unknown constant k, then we read the other probabilities from the table.
Step 2: List the probabilities.
For \(x_i=0\): \(P=0.2\).
For \(x_i=1\): \(P=k(1)=k\).
For \(x_i=2\): \(P=k(2)=2k\).
For \(x_i=3\): \(P=k(4-3)=k\).
Step 3: Find k (match A).
The total probability is 1: \[ 0.2 + k + 2k + k = 1 \] \[ 4k = 0.8 \] \[ k = 0.2 \] So A matches III (0.2).
Step 4: Match B.
At least two hours means \(X \geq 2\): \[ P(X \geq 2) = P(2) + P(3) = 2k + k = 3k = 0.6 \] So B matches IV (0.6).
Step 5: Match C.
At most three hours means every possible value 0, 1, 2 and 3, because X cannot take any other value. So the probability is the total, which is 1. So C matches I (1).
Step 6: Match D.
At most one hour means X is 0 or 1: \[ P(0) + P(1) = 0.2 + k = 0.2 + 0.2 = 0.4 \] So D matches II (0.4).
Step 7: Choose the option.
We have A-III, B-IV, C-I, D-II. This is option 2. Option 1 has B-II and D-IV, which is swapped. Options 3 and 4 give A-IV, but k is 0.2, not 0.6, so they are wrong.
Final Answer:
The matching is A-III, B-IV, C-I, D-II. \[ \boxed{\text{Option (2)}} \]
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