Two cards are drawn without replacement from a standard deck of 52 cards. Find the probability a) both cards are red ,b) both cards are the same color, c) the second card is a king given that the first card is a queen, d) the second card is the queen of hearts given that the first card is black
prob both red = (26/52)(25/51) = 25/102
prob both same colour
= 25/102 + 25/102 = 50/102 = 25/51
prob (queen then king) = (4/52)(4/51) = 4/663
prob( black then queen of hearts) = (26/52)(1/51) = 1/102
a) The probability of drawing a red card on the first draw is 26/52 since there are 26 red cards in a standard deck. After one red card is removed, there are 25 red cards left out of 51 cards. So the probability of drawing a red card on the second draw is 25/51. To find the probability of both cards being red, we multiply the probabilities of each draw: (26/52) * (25/51) = 25/102.
b) To find the probability of both cards being the same color, we need to consider two cases: both cards being red, and both cards being black. From part a), we already know the probability of both cards being red is 25/102. The probability of both cards being black can be calculated in a similar manner: (26/52) * (25/51) = 25/102.
Adding these two probabilities together gives us the overall probability of both cards being the same color: 25/102 + 25/102 = 50/102 = 25/51.
c) Given that the first card is a queen, there are 51 cards left in the deck, and 4 of them are kings. So the probability of drawing a king on the second card, given that the first card is a queen, is 4/51.
d) Given that the first card is black, there are 26 black cards left in the deck, including the queen of hearts. Since we know the first card is black, the probability of drawing the queen of hearts on the second card is 1/26.
To find the probability of each event, we need to first calculate the total number of possible outcomes and the number of favorable outcomes for each scenario.
a) Both cards are red:
There are 26 red cards in a standard deck. After drawing the first card, there are 25 red cards remaining for the second draw. So, the number of favorable outcomes is (26 * 25) and the total number of possible outcomes is (52 * 51) since we draw the cards without replacement. Therefore, the probability that both cards are red is:
P(both cards are red) = (26/52) * (25/51) = 0.2451 or approximately 24.51%.
b) Both cards are the same color:
There are two possibilities for this event to occur: both cards are red or both cards are black. We already found the probability that both cards are red to be 0.2451, so the probability that both cards are black is the same. Therefore, the probability that both cards are the same color is:
P(both cards are the same color) = P(both cards are red) + P(both cards are black) = 0.2451 + 0.2451 = 0.4902 or approximately 49.02%.
c) The second card is a king given that the first card is a queen:
After drawing the first card, there are 51 cards remaining in the deck. Since the first card is a queen, there are 4 kings remaining in the deck. So, the number of favorable outcomes is 4 and the total number of possible outcomes is 51. Therefore, the probability that the second card is a king given that the first card is a queen is:
P(second card is a king | first card is a queen) = 4/51 = approximately 0.0784 or 7.84%.
d) The second card is the queen of hearts given that the first card is black:
After drawing the first card, there are 51 cards remaining in the deck. Since the first card is black, there are 25 black cards remaining in the deck. The queen of hearts is also black, so there is only 1 favorable outcome. Therefore, the probability that the second card is the queen of hearts given that the first card is black is:
P(second card is the queen of hearts | first card is black) = 1/51 = approximately 0.0196 or 1.96%.
To find the probability of an event, we need to know the total number of possible outcomes and the number of favorable outcomes.
a) To find the probability of both cards being red, we need to determine the total number of red cards and the total number of cards in the deck. There are 26 red cards in a standard deck (13 hearts and 13 diamonds), and the total number of cards is 52. Since we are drawing cards without replacement, after selecting the first red card, there will be 25 red cards left and 51 cards remaining in total. The probability of drawing a second red card is (25/51). Thus, the probability of drawing two red cards is:
P(both cards are red) = (26/52) * (25/51) = 0.245 or 24.5%
b) To find the probability of both cards being the same color, we consider the possibilities of both red cards or both black cards. The probability of both cards being red is (26/52) * (25/51), as explained in part a. Similarly, the probability of drawing two black cards without replacement is (26/52) * (25/51). Therefore, the probability of both cards being the same color is:
P(both cards are the same color) = P(both red) + P(both black) = 0.245 + 0.245 = 0.49 or 49%
c) To find the probability of the second card being a king given that the first card is a queen, we need to consider the reduced deck after drawing a queen as the first card. We know that a standard deck contains 4 queens, and after drawing one, there will be 51 cards remaining. Out of these 51 cards, there are 4 kings. Therefore, the probability of drawing a king as the second card, given that the first card is a queen, is:
P(second card is a king | first card is a queen) = (4/51) = 0.078 or 7.8%
d) To find the probability of the second card being the queen of hearts given that the first card is black, we need to consider the reduced deck only containing black cards after drawing the first card. Since there are 26 black cards in a standard deck, after drawing one, there will be 25 black cards remaining. Out of these 25 cards, only one is the queen of hearts. Therefore, the probability of drawing the queen of hearts as the second card, given that the first card is black, is:
P(second card is queen of hearts | first card is black) = (1/25) = 0.04 or 4%