Question

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

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Jayne

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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}

$$.

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{51}{221}

$$.

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