Two cards are drawn at random from a standard deck of $52$ cards and not replaced.

Question

Two cards are drawn at random from a standard deck of $52$ cards and not replaced. What is the probability that the first card is a face card and the second card is a seven? What is the answer to this maths problem and how is the answer done?

Answer 1

There 12 face cards, so the probability that the first card is a face card is $\frac{12}{52}$. Then there are 51 cards left. If the first card was a face card, then 4 sevens are still in the deck. So the probability that the first card is a face card and the second card a seven is $\frac{12}{52} \times \frac{4}{51} = \frac{4}{221}$.

Answer 2

There are 12 face cards in a deck of 52 cards. Hence, probability that the first card drawn is face card is 12/52. Now, the cards drawn are not replaced. Hence we are now left with 51 cards in the deck. There are 4 cards of seven in the deck. Hence the probability for second card to be 7 is 4/51.

Hence the total probability that the first card is a face card AND the second card is a seven is

(12/52) * (4/51) = (4/221)