I'm looking for the probability that we first get a king on the nth card draw when drawing from a pack of 52 cards.
Here's what I have done -
Let $A_i$ be the event that we don't get a king on card $i$.
Let $K$ be the event of getting a king.
Want to find
P(King occurs on nth card)
$= P(A_1 \cap A_2 \cap ... \cap A_{n-1} \cap K)$
$= P(K | A_1 \cap A_2 \cap ... \cap A_{n-1})P(A_{n-1}| A_1 \cap A_2 \cap ... \cap A_{n-2})...P(A_2 | A_1)P(A_1)$
$= \frac{48}{52}\frac{48}{51}\frac{48}{50}...\frac{48}{52-n+2}\frac{4}{52-n+1}$
$= \frac{(4)(48)^{52-n+2}}{^{52} P_n}$
Is that correct?