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.
. Two cards are drawn at random, without replacement, from a standard 52-card deck. Find the
probability that:
(a) both cards are the same color
(b) the first card is a face card and the second card is black
To find the probabilities for each given situation, we first need to determine the sample space and the favorable outcomes for each event.
In this case, we are drawing two cards without replacement from a standard deck of 52 cards. Without replacement means that once a card is drawn, it is not put back into the deck before drawing the next card.
a) Probability that both cards are red:
There are 26 red cards in a standard deck (13 hearts and 13 diamonds). The probability of drawing a red card on the first draw is 26/52 (since there are 52 cards total in the deck). After the first red card is drawn, there are 25 red cards left in a deck of 51 cards. So, the probability of drawing a second red card is 25/51. To find the probability that both cards are red, we multiply the probabilities of each event: (26/52) * (25/51) = 1/2 * 25/51 = 25/102.
b) Probability that both cards are the same color:
Since we are drawing without replacement, the color of the first card will affect the probability of the second card.
If the first card is red, then the probability of drawing a second red card (both are red) is 25/51 (as found in part a).
If the first card is black, then the probability of drawing a second black card (both are black) is 26/51.
Therefore, the probability that both cards are the same color is the sum of these two probabilities: 25/51 + 26/51 = 51/51 = 1.
c) Probability that the second card is a king given that the first card is a queen:
If the first card is a queen, then there are 3 queens left in the deck (since one queen is already drawn), and there are 51 cards remaining. So, the probability of drawing a king as the second card given that the first card is a queen is 4/51.
d) Probability that the second card is the queen of hearts given that the first card is black:
If the first card is black, then there are 26 black cards in the deck (13 spades and 13 clubs). Out of these 26 black cards, there is only one queen of hearts. So, the probability of drawing the queen of hearts given that the first card is black is 1/26.
To summarize:
a) The probability that both cards are red is 25/102.
b) The probability that both cards are the same color is 1.
c) The probability that the second card is a king given that the first card is a queen is 4/51.
d) The probability that the second card is the queen of hearts given that the first card is black is 1/26.