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
To find the probabilities, we need to understand the concept of probability and the counting principles involved.
a) To find the probability that both cards are red, we need to determine the number of favorable outcomes (red cards) and divide it by the number of total possible outcomes.
There are 26 red cards in a standard deck of 52 cards. When the first card is drawn, there are 26 red cards and 51 cards remaining. When the second card is drawn, there are 25 red cards and 50 cards remaining. Therefore, the probability that both cards are red is (26/52) * (25/51) = 0.2451 or approximately 24.51%.
b) To find the probability that both cards are the same color, we can consider two cases: both red or both black cards. We calculate the probability for each case and then add them together.
Case 1: Both red cards.
Using the same reasoning as in part a, the probability of drawing two red cards is (26/52) * (25/51) = 0.2451.
Case 2: Both black cards.
When the first card is drawn, there are 26 red cards and 26 black cards remaining. Therefore, the probability of drawing a black card on the second draw is (26/52) * (25/51) = 0.2451.
Now, we add the probabilities of both cases:
0.2451 + 0.2451 = 0.4902 or approximately 49.02%.
Therefore, the probability that both cards are the same color is approximately 49.02%.
c) To find the probability that the second card is a king given that the first card is a queen, we know that the first card is a queen, leaving us with 51 cards. There are 4 kings remaining in the deck.
Therefore, the probability that the second card is a king, given that the first card is a queen, is 4/51.
d) To find the probability that the second card is the queen of hearts given that the first card is black, we know that the first card is black, leaving us with 26 black cards remaining. Among those, only one is the queen of hearts.
Therefore, the probability that the second card is the queen of hearts, given that the first card is black, is 1/26.
To find the probabilities in each scenario, we need to determine the number of favorable outcomes and the total number of possible outcomes.
a) Probability that both cards are red:
In a standard deck, there are 26 red cards (13 hearts and 13 diamonds), so the first favorable outcome is selecting a red card. After drawing one red card, there are 25 red cards remaining out of 51 total cards. Therefore, the probability of drawing a second red card is:
P(both cards are red) = (26/52) * (25/51) = 0.2451
b) Probability that both cards are the same color:
Since we have already calculated the probability of drawing both red cards, we can also calculate the probability of drawing both black cards. In a standard deck, there are 26 black cards (13 clubs and 13 spades). So, the probability of drawing both black cards is:
P(both cards are black) = (26/52) * (25/51) = 0.2451
Now, we can calculate the probability of drawing two cards of the same color by adding the probabilities of picking both red cards and both black cards:
P(both cards are the same color) = P(both cards are red) + P(both cards are black) = 0.2451 + 0.2451 = 0.4902
c) Probability that the second card is a king given that the first card is a queen:
Since we are given that the first card is a queen from a standard deck of 52 cards, there are 51 remaining cards. Out of the 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) ≈ 0.0784
d) Probability that the second card is the queen of hearts given that the first card is black:
Since we are given that the first card drawn is black (clubs or spades), there are 26 remaining black cards out of the total of 51 cards. Out of these 26 black cards, there is only one queen, which 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 the queen of hearts | first card is black) = (1/26) ≈ 0.0385
Note: For questions (c) and (d), the probabilities are independent of the first card once it has been drawn.