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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## Answer to a math 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

Jayne
4.4
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}
.

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