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
The probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
a) 26/52 * (26-1)/(52-1) = ?
b) Either-or probabilities are found by adding the individual probabilities. The probability above for red plus the same probability for black.
c) 4/52 * 4/(52-1) = ?
Use a similar thinking process for d.
To find the probabilities in this scenario, we need to understand the concept of conditional probability and the counting principle.
a) Probability that both cards are red:
In a standard deck of 52 cards, half of the cards are red (26 out of 52). When drawing the first card, the probability of picking a red card is 26/52. Now, when drawing the second card without replacement, there will be one less card in the deck, so the probability of drawing another red card is now 25/51. To find the probability of both events occurring, we multiply the probabilities together:
P(both cards are red) = (26/52) * (25/51)
b) Probability that both cards are the same color:
Again, half of the cards are red and the other half are black. We already calculated that the probability of drawing a red card on the first draw is 26/52. Now, for the second draw, if the first card drawn is red, then there are 25 red cards remaining out of 51 cards. If the first card drawn is black, then there are 26 black cards remaining out of 51 cards. So, the probability of drawing a card of the same color is the sum of these two cases:
P(both cards are the same color) = (26/52) * (25/51) + (26/52) * (25/51)
c) Probability that the second card is a king given that the first card is a queen:
When the first card is a queen, there are 51 cards left in the deck, including 3 kings. Therefore, the probability of drawing a king as the second card when the first card is a queen is 3/51. In this case, we do not need to multiply by the probability of drawing a queen since it was given in the condition:
P(second card is a king | first card is a queen) = 3/51
d) Probability that the second card is the queen of hearts given that the first card is black:
When the first card is black, there are 26 black cards left in the deck. The queen of hearts is only one card out of those 26 black cards. Therefore, the probability of drawing the queen of hearts as the second card when the first card is black is 1/26. Again, in this case, we do not need to multiply by the probability of drawing a black card since it was given in the condition:
P(second card is the queen of hearts | first card is black) = 1/26
Please note that the probabilities may vary if the deck is not a standard deck or if there are any specific rules or conditions that were not mentioned in the question.