Question

compute the probability of drawing two clubs from a standard deck of 52 cards. round your answer to four decimal places.
answer:
compute the probability of drawing two face cards from a standard deck of 52 cards. round your answer to four decimal places.
answer:
compute the probability of drawing two aces from a standard deck of 52 cards. round your answer to four decimal places.
answer:

Explanation:

Step1: Calculate probability of first - draw

The probability of drawing a club on the first draw from a standard deck of 52 cards is the number of clubs divided by the total number of cards. There are 13 clubs in a 52 - card deck, so $P_1=\frac{13}{52}=\frac{1}{4}$.

Step2: Calculate probability of second - draw

After drawing one club on the first draw, there are 12 clubs left and 51 cards left in the deck. So the probability of drawing a club on the second draw given that a club was drawn on the first draw is $P_2=\frac{12}{51}$.

Step3: Calculate joint probability

By the multiplication rule for independent events (in the case of sequential draws without replacement), the probability of drawing two clubs is $P = P_1\times P_2=\frac{13}{52}\times\frac{12}{51}=\frac{13\times12}{52\times51}=\frac{156}{2652}\approx0.0588$.

Step4: Calculate probability of drawing two face - cards (first draw)

There are 12 face - cards (4 Jacks, 4 Queens, and 4 Kings) in a 52 - card deck. So the probability of drawing a face - card on the first draw is $P_{1f}=\frac{12}{52}=\frac{3}{13}$.

Step5: Calculate probability of drawing two face - cards (second draw)

After drawing one face - card on the first draw, there are 11 face - cards left and 51 cards left in the deck. So the probability of drawing a face - card on the second draw given that a face - card was drawn on the first draw is $P_{2f}=\frac{11}{51}$.

Step6: Calculate joint probability of face - cards

The probability of drawing two face - cards is $P_f=P_{1f}\times P_{2f}=\frac{12}{52}\times\frac{11}{51}=\frac{12\times11}{52\times51}=\frac{132}{2652}\approx0.0498$.

Step7: Calculate probability of drawing two aces (first draw)

There are 4 aces in a 52 - card deck. So the probability of drawing an ace on the first draw is $P_{1a}=\frac{4}{52}=\frac{1}{13}$.

Step8: Calculate probability of drawing two aces (second draw)

After drawing one ace on the first draw, there are 3 aces left and 51 cards left in the deck. So the probability of drawing an ace on the second draw given that an ace was drawn on the first draw is $P_{2a}=\frac{3}{51}$.

Step9: Calculate joint probability of aces

The probability of drawing two aces is $P_a=P_{1a}\times P_{2a}=\frac{4}{52}\times\frac{3}{51}=\frac{4\times3}{52\times51}=\frac{12}{2652}\approx0.0045$.

Answer:

0.0588
0.0498
0.0045