conditional probability and the multiplication rule
1. two cards are selected in sequence from a standard deck (52 cards). find the probability that the
second card is an ace, given that the first card is:
a) a seven (assume that it is not replaced):
b) an ace (assume that it is not replaced):
c) what would be the answer in a) and b) if the first card is replaced?

If the first card (not replaced) is a 7, you're left with 51 cards, and the four aces are all there: you have probability 4/51 of picking an ace.

If the first card (not replaced) is an ace, you're left again with 51 cards, but these time only 3 aces are remaining in the pile: you have probability 3/51 of picking an ace.

Were the cards replaced, the information about the first card drawn would be completely meaningless, since the second pick would be a pick from a standard deck, and we'd have probability 4/52=1/13 of picking an ace.

In other words, if cards are replaced, it is like every pick is the first one, because whatever happened with the first pick has no consequences on the second.

On the other hand, if we do not replace cards, the outcome of the first pick is important, because we know that the card we picked will not be in the deck anymore.