# Sampling without replacement while avoiding an element

Let $$p$$ be a distribution over $$N$$ objects, pick an object $$k$$ from $$N$$, and define $$p^*(x)$$ to be 0 if $$x = k$$ and $$p(x) / (1 - p(k))$$ otherwise.

Suppose I want to sample $$n$$ items from $$p^*$$ without replacement (that is, every time I draw an element, I set it aside and renormalise the probabilities of the remaining items), but I can only draw samples WOR from $$p$$. Is the following a valid sampling WOR scheme for $$p^*$$:

1. Draw $$n+1$$ items from $$p$$ WOR.
2. If any of the $$n+1$$ items are equal to $$k$$, remove it from the sample.
3. Otherwise take the first $$n$$ items.

Consider drawing $$N$$ items from $$p$$ WOR. This is a random ordering of your items. Picking the $$n$$ first items from this list is equivalent to drawing $$n$$ items from $$p$$.
Now consider the effect of your steps 2 and 3 of this interpretation. 2 and 3 together is equivalent to removing items $$k$$ from your list of $$N$$ randomly ordered items and picking the remaining $$n$$ first items is equivalent to drawing $$n$$ items from the set of size $$N-N_k$$ where $$N_k$$ items $$k$$ have been removed in accordance with $$p^*$$.