Suppose the two joker cards are left in a standard deck of cards. One joker is red and the other is black. A single card is drawn from the deck of 54 cards, returned to the deck, and a second card is drawn. Determine the probability of drawing a red joker or a red ace on either draw.
So there are two red aces and one red joker.
Instinctively, I would think $\frac{3}{54}\times\frac{3}{54}=\frac{1}{324} $
But 1 - Pr(no red joker or red ace on either draw) = $1-\frac{51}{54}\times\frac{51}{54}=\frac{35}{324} $
I can't figure this out...
The question asks you to determine the probability of drawing a red joker or a red ace on either draw. So your second attempt using complimentary method is correct. In your first approach you are calculating the probability that both draws are either a red ace or a red joker. The correct answer would be,
$ \displaystyle \frac{3}{54} \cdot \frac{51}{54} + \frac{51}{54} \cdot \frac{3}{54} + \frac{3}{54} \cdot \frac{3}{54} = \frac{35}{324}$