Two cards are drawn without replacement from an ordinary deck Find probability that the first is a face card, given that the second is not a face card

Solution:
1. Define the events:
- Let \( A \) be the event that the first card drawn is a face card.
- Let \( B \) be the event that the second card drawn is not a face card.

2. Determine the total number of face cards and non-face cards:
- Total face cards: 12 (Jack, Queen, King in each of 4 suits: 3 * 4 = 12)
- Total non-face cards: 52 - 12 = 40

3. Use the conditional probability formula \( P(A|B) = \\frac{P(A \\cap B)}{P(B)} \):
- Calculate \( P(B) \):
* Number of favorable outcomes for \( B \) in the second draw = 40 (since it is not a face card).
* Possible outcomes of the second card being not a face card given any first card: 52 (total cards initially).
* Hence, \( P(B) = \\frac{40}{51} \\).

- Calculate \( P(A \\cap B) \):
* Number of favorable outcomes: First card is a face card = 12.
* Second card is not a face card = 40.
* Probability of first card being a face card = \\frac{12}{52} = \\frac{3}{13} \\.
* Probability of the second card being not a face card given the first was a face card = \\frac{40}{51} \\.
* Hence, \( P(A \\cap B) = \\frac{12}{52} \\cdot \\frac{40}{51} = \\frac{480}{2652} = \\frac{40}{221} \\).

4. Calculate \( P(A|B) \):
* Substituting into the formula gives:
* P(A|B) = \frac{\frac{40}{221}}{\frac{40}{51}} = \frac{40}{221} \times \frac{51}{40} = \frac{51}{221} \). The Conditional Probability is:
P(A|B) = \\frac{51}{221}
$$.