Find the probability of drawing a face card that is a Diamond on the first draw, replacing it and drawing a black card on the second draw.

3/52 * 13/52

To find the probability of drawing a face card that is a Diamond on the first draw, replacing it, and drawing a black card on the second draw, we need to break down the problem step-by-step.

Step 1: Determine the number of favorable outcomes for drawing a face card that is a Diamond on the first draw.
In a standard deck of 52 cards, there are a total of 4 face cards (Jacks, Queens, and Kings) for each suit: Diamond, Heart, Club, and Spade. Out of these 4 face cards for each suit, only 1 of them is a Diamond (the Jack of Diamonds, Queen of Diamonds, or King of Diamonds). So, there are 3 favorable outcomes for drawing a face card that is a Diamond on the first draw.

Step 2: Determine the total number of possible outcomes for drawing a card on the first draw.
In a standard deck of 52 cards, there are a total of 52 cards.

Step 3: Determine the probability of drawing a face card that is a Diamond on the first draw.
The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 3 / 52

Step 4: Determine the number of favorable outcomes for drawing a black card on the second draw.
In a standard deck of 52 cards, there are 2 black suits: Club and Spade. Each of these suits has 9 numbered cards (2, 3, 4, 5, 6, 7, 8, 9, and 10) and 3 face cards (Jack, Queen, and King) that are black. So, there are a total of 9 + 3 = 12 favorable outcomes for drawing a black card on the second draw.

Step 5: Determine the total number of possible outcomes for drawing a card on the second draw.
Since the first card drawn in Step 1 is replaced, the total number of cards in the deck remains the same at 52 cards.

Step 6: Determine the probability of drawing a black card on the second draw.
The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 12 / 52

Step 7: Combine the probabilities from Step 3 and Step 6.
Since the two events (drawing a face card that is a Diamond on the first draw and drawing a black card on the second draw) are independent, we multiply the probabilities together to find the overall probability:
Overall Probability = Probability of drawing a face card that is a Diamond on the first draw × Probability of drawing a black card on the second draw
Overall Probability = (3 / 52) × (12 / 52)
Overall Probability = 36 / 2704

Thus, the probability of drawing a face card that is a Diamond on the first draw and then replacing it and drawing a black card on the second draw is 36/2704.

To find the probability of drawing a face card that is a Diamond on the first draw, replacing it, and then drawing a black card on the second draw, we need to consider the following:

1. Number of face cards: In a standard deck of cards, there are 12 face cards (King, Queen, and Jack) in total. However, there are only 3 face cards that are Diamonds (Diamond King, Diamond Queen, and Diamond Jack).

2. Total number of cards: A standard deck of cards consists of 52 cards.

Now, let's break down the calculation step by step:

Step 1: Probability of drawing a face card that is a Diamond on the first draw.
The probability of drawing a Diamond face card on the first draw is calculated by dividing the number of favorable outcomes (3 Diamond face cards) by the number of possible outcomes (52 cards in total). Therefore, the probability is 3/52.

Step 2: Replacing the card.
Since we are replacing the card after the first draw, the deck will be reshuffled, and the total number of cards remains the same at 52.

Step 3: Probability of drawing a black card on the second draw.
In a standard deck, there are 26 black cards (13 Spades and 13 Clubs). The probability of drawing a black card on the second draw, given that we already drew a Diamond face card and replaced it, is 26/52. Notice that we are using 52 total cards because we replaced the Diamond face card back into the deck.

Step 4: Multiplying the probabilities.
To calculate the combined probability of both events happening (drawing a Diamond face card and replacing it, then drawing a black card), we multiply the probabilities from step 1 and step 3:
(3/52) * (26/52) = 1/4 * 1/2 = 1/8.

Therefore, the probability of drawing a face card that is a Diamond on the first draw, replacing it, and drawing a black card on the second draw is 1/8 or 0.125 (12.5%).