Question

In Simple Random Sampling Without Replacement (SRSWOR), what is the probability of selecting a specific unit at the second draw?

• A. \( \dfrac{1}{N} \)
• B. \( \dfrac{1}{N-1} \)
• C. \( \dfrac{2}{N} \)
• D. \( \dfrac{1}{N^2} \)

Hint:

In SRSWOR, every unit has an equal chance of appearing in any position of the sample. Thus, the probability that a specific unit appears in the \(k^{th}\) draw is always: \[ \frac{1}{N} \] where \(N\) is the population size.

Answer 1

The Correct Option is A

Concept:
In Simple Random Sampling Without Replacement (SRSWOR), each unit in the population has an equal probability of being selected at any draw. Even though units are not replaced, the probability that any particular unit appears in a specific draw remains the same due to symmetry of selection. If the population size is \(N\), then the probability that a specific unit is selected in any particular position of the sample is: \[ P = \frac{1}{N} \]
Step 1: Consider the population size.
Let the population contain \(N\) units, and suppose we are interested in the probability that a particular unit (say unit \(i\)) is selected at the second draw.
Step 2: Consider the two possible ways this can happen.
The specific unit can appear in the second draw if:

• It was not selected in the first draw.
• It is selected in the second draw.

Thus, \[ P(\text{unit } i \text{ at second draw}) = P(\text{not selected in first}) \times P(\text{selected in second}) \] \[ = \left(\frac{N-1}{N}\right) \left(\frac{1}{N-1}\right) \] \[ = \frac{1}{N} \]
Step 3: Interpret the result.
Hence, the probability that a particular unit is selected at the second draw is: \[ \frac{1}{N} \] \[ \therefore \text{The correct answer is } \dfrac{1}{N}. \]