Two cards are drawn from a standard deck of 52 cards, without replacement.  Determine the following probabilities.  Show your work and round answers to three decimal places.

a. P(both cards are face cards) = ____

b. P(second card is a face card given that the first card was not a face card) = ______

Two cards are drawn from a standard deck of 52 cards, without replacement.  Determine the following probabilities.  Show your work and round answers to three decimal places.

a. P(both cards are face cards) = ____

b. P(second card is a face card given that the first card was not a face card) = ______

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To determine the probabilities, let's first outline the relevant facts about a standard deck of 52 cards:

- There are 12 face cards in total (3 face cards in each of the 4 suits: Jack, Queen, King).
- The remaining cards are non-face cards (52 - 12 = 40).

### Part (a): P(both cards are face cards)

1. **Calculate the probability of drawing a face card first:**
   - There are 12 face cards in a 52-card deck.
   - Probability of drawing a face card first: 
     \[
     P(\text{1st face card}) = \frac{12}{52}
     \]

2. **Calculate the probability of drawing a face card second (after one face card has already been drawn):**
   - After drawing one face card, there are now 11 face cards left and 51 total cards.
   - Probability of drawing a face card second:
     \[
     P(\text{2nd face card} | \text{1st face card}) = \frac{11}{51}
     \]

3. **Combine these probabilities:**
   \[
   P(\text{both face cards}) = P(\text{1st face card}) \times P(\text{2nd face card} | \text{1st face card}) = \frac{12}{52} \times \frac{11}{51}
   \]

4. **Calculate the final probability:**
   \[
   P(\text{both face cards}) = \frac{12}{52} \times \frac{11}{51} = \frac{132}{2652} \approx 0.050
   \]

### Part (b): P(second card is a face card given that the first card was not a face card)

1. **Calculate the probability that the first card is not a face card:**
   - There are 40 non-face cards.
   - Probability of drawing a non-face card first:
     \[
     P(\text{1st non-face card}) = \frac{40}{52}
     \]

2. **Calculate the probability of drawing a face card second (after a non-face card has been drawn):**
   - The number of face cards remains the same (12), but the total number of cards is now 51.
   - Probability of drawing a face card second:
     \[
     P(\text{2nd face card} | \text{1st non-face card}) = \frac{12}{51}
     \]

3. **Combine these probabilities using the conditional probability:**
   \[
   P(\text{2nd face card} | \text{1st non-face card}) = \frac{12}{51} \text{ (since we only need this part)}
   \]

4. **Calculate the final probability:**
   \[
   P(\text{2nd face card | 1st non-face}) = \frac{12}{51} \approx 0.235
   \]

### Final Answers

a. \( P(\text{both cards are face cards}) = 0.050 \)

b. \( P(\text{second card is a face card | first card was not a face card}) = 0.235 \)

If you need any further details or have questions, feel free to ask! Here are some related questions you might consider:

1. What is the probability of drawing at least one face card in two draws?
2. How does the probability change if we draw with replacement?
3. What is the probability of drawing two cards of the same suit?
4. How can we generalize these probabilities for larger or smaller decks?
5. What other card combinations could you analyze with similar methods? 

**Tip:** Remember to always consider the changing total number of cards and favorable outcomes when dealing with probabilities without replacement!

Two cards are drawn from a standard deck of 52 cards, without replacement. Determine the following probabilities: a. P(both cards are face cards) = ____ b. P(second card is a face card given that the first card was not a face card) = ______

Explore the probabilities of drawing two face cards from a standard deck of 52 cards. Learn to calculate P(both cards are face cards) and P(second card is a face card given the first was not a face card), complete with detailed solutions and examples. Suitable for Grades 9-12.

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