Discussion
Consider a population of size N = 6 as {1, 2, 3, 4, 5, 6} with their auxiliary variable values as :
\(x_{1} = 100\), \(x_{2} = 200\), \(x_{3} = 300\), \(x_{4} = 400\), \(x_{5} = 500\) and \(x_{6} = 500\).
We want to draw a random sample of n = 3 using PPSWOR design. Let \(U_{r}\) denote the unit selected at the rth draw for r = 1, 2, 3.
Then :
\[ P[U_{3} = 2 \mid U_{1} = 5, U_{2} = 3] = \]
Answer: A
The probability is \(\frac{1}{6}\).
In PPSWOR each draw selects among the remaining units with probability proportional to their sizes. The total size is \(100 + 200 + 300 + 400 + 500 + 500 = 2000\). After units 5 (size 500) and 3 (size 300) are drawn, the remaining total is
\[ 2000 - 500 - 300 = 1200 \]
so
\[ \begin{aligned} P[U_{3} = 2 \mid U_{1} = 5, U_{2} = 3] &= \frac{200}{1200} \\ &= \frac{1}{6} \end{aligned} \]
\(\frac{1}{10}\) is the unconditional first-draw probability \(\frac{200}{2000}\), which ignores the units already removed. \(\frac{2}{11}\) and \(\frac{2}{13}\) remove the wrong sizes from the total.
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