. 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
1/2
To find the probability in each case, we need to first determine the total number of possible outcomes and the number of favorable outcomes.
(a) Probability that both cards are the same color:
First, let's determine the total number of possible outcomes. In a standard 52-card deck, there are 52 cards.
Now, let's find the number of favorable outcomes, i.e., the number of ways we can choose two cards of the same color. We have two cases: both cards are red (either heart or diamond) or both cards are black (either club or spade).
Case 1: Both cards are red: In a deck of 52 cards, 26 cards are red. We need to choose 2 out of the 26 red cards. This can be done in (26 choose 2) or C(26, 2) ways.
Case 2: Both cards are black: In a deck of 52 cards, 26 cards are black. We need to choose 2 out of the 26 black cards. Again, this can be done in (26 choose 2) or C(26, 2) ways.
The total number of favorable outcomes is the sum of the favorable outcomes from each case.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
P(both cards are the same color) = (number of favorable outcomes) / (total number of possible outcomes)
= (favorable outcomes of case 1 + favorable outcomes of case 2) / 52C2
(b) Probability that the first card is a face card and the second card is black:
First, let's determine the total number of possible outcomes. In a standard 52-card deck, there are 52 cards.
To find the number of favorable outcomes, we need to consider the two conditions: the first card is a face card (Jack, Queen, or King) and the second card is black (club or spade).
Case 1: The first card is a face card: In a deck of 52 cards, there are 12 face cards (3 face cards for each suit - Jack, Queen, and King). We can choose any of these 12 face cards as the first card.
Case 2: The second card is black: In a deck of 52 cards, 26 cards are black. After drawing the first card, there are 51 cards remaining, out of which 26 are black. We need to choose 1 out of these 26 black cards as the second card.
The total number of favorable outcomes is the product of the favorable outcomes from each case.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
P(first card is a face card and second card is black) = (number of favorable outcomes) / (total number of possible outcomes)
= (favorable outcomes of case 1 * favorable outcomes of case 2) / 52C2
To get the numerical values, you can calculate the combinations using the formula: C(n, r) = n! / (r!(n-r)!), where n is the total number of items, and r is the number of items chosen.