In a card deck, what is the probability that choosing 5 cards will include at least 1 pair

Question

In a card deck (52 cards), what is the probability that choosing 5 cards will include at least 1 pair? Triples or still counted as pairs. It's a silly question but I just can't seem to get it.

Answer 1

Note that your question is equivalent to asking what 1 - (the probability that you will choose 0 cards with the same numerical value) is, since you either get at least one pair or you get no pairs. This is a much easier calculation than considering each possible scenario (i.e. exactly one pair, exactly 2 pairs, 3 of a kind, etc.). You can model this problem by thinking about picking five random cards from the deck. Obviously, the first card does not have the "avoiding pairs" restriction on it because there are no other cards that it could be paired with yet. Therefore, you can pick any one of the 52 cards, which we denote by $\frac{52}{52}$. Now there are 51 cards left, and you want to avoid picking the same numerical value as you did before. Therefore, you need to avoid 3 out of the remaining 51, or inversely, you can pick $\frac{48}{51}$. Now you just need to continue this method with the other 3 cards and subtract your result from 1 because of what I mentioned above.