Two cards are drawn at random from an ordinary deck of 52 playing cards. What is the probability of getting two faces if:
a) the first card is replaced before the second card is drawn?
b) the first card is not replaced before the second card is drawn?
Since there are 12 face cards,
(a) 12/52 * 12/52
(b) 12/52 * 11/51
a) To find the probability of getting two faces when the first card is replaced before the second card is drawn, we need to determine how many favorable outcomes there are and divide it by the total number of possible outcomes.
1. Determine the number of favorable outcomes:
There are 12 face cards in a standard deck, with 3 face cards in each of the four suits (Jacks, Queens, and Kings). So, the number of ways to select two face cards out of 12 is given by the combination formula: C(12, 2) = 66.
2. Determine the total number of possible outcomes:
Since the first card is replaced before drawing the second card, the total number of possible outcomes for the first and second cards are both 52 (since each card is replaced).
3. Calculate the probability:
The probability is given by the number of favorable outcomes divided by the total number of possible outcomes:
P(a) = favorable outcomes / total outcomes = 66 / (52 * 52) = 66 / 2704 ≈ 0.0244 or 2.44%.
b) To find the probability of getting two faces when the first card is not replaced before the second card is drawn, we follow the same steps as above but with a slight adjustment.
1. Determine the number of favorable outcomes:
Since the first card is not replaced, there will be one less card in the deck for the second draw. So, the number of ways to select two face cards from the remaining 51 cards is given by the combination formula: C(12, 2) = 66.
2. Determine the total number of possible outcomes:
The first card can be any of the 52 cards, and the second card can be any of the remaining 51 cards.
3. Calculate the probability:
P(b) = favorable outcomes / total outcomes = 66 / (52 * 51) ≈ 0.123 or 12.3%.
Therefore, the probabilities are:
a) P(a) ≈ 0.0244 or 2.44%
b) P(b) ≈ 0.123 or 12.3%
a) If the first card is replaced before the second card is drawn, then the probability of getting a face card (there are 12 face cards in a deck) on each draw is the same.
The probability of drawing a face card on the first draw is 12/52.
Since the first card is replaced before the second card is drawn, the probability of drawing a face card on the second draw is also 12/52.
To find the probability of both events occurring, we multiply the probabilities together:
P(Two Faces with Replacement) = (12/52) * (12/52) = 144/2704 ≈ 0.053, or about 5.3%
b) If the first card is not replaced before the second card is drawn, then the probability of getting a face card changes for the second draw.
After the first card is drawn, there are now 51 cards remaining, with 11 face cards left.
The probability of drawing a face card on the first draw is still 12/52, but the probability on the second draw becomes 11/51.
To find the probability of both events occurring, we multiply the probabilities together:
P(Two Faces without Replacement) = (12/52) * (11/51) ≈ 0.052, or about 5.2%