Cards are drawn one-by-one without replacement from a well shuffled pack of 52 cards. Then the probability that a face card ( Jack, Queen or King) will appear for the first time on the third turn is equal to

Options

A. `(300)/(2197)`
B. `(36)/(85)`
C. `(12)/(85)`
D. `(4)/(51)`

Answer

To find the probability that a face card (Jack, Queen, or King) will appear for the first time on the third turn when drawing cards one-by-one without replacement from a well-shuffled pack of 52 cards, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Cards and Face Cards:**
- A standard deck has 52 cards.
- There are 12 face cards (3 face cards: Jack, Queen, King from each of the 4 suits).
- Therefore, the number of non-face cards is \( 52 - 12 = 40 \).

2. **Determine the Required Outcome:**
- We want the first two cards drawn to be non-face cards, and the third card drawn to be a face card.

3. **Calculate the Probability of Drawing Non-Face Cards First:**
- For the first card, the probability of drawing a non-face card is:
\[ P(\text{1st card is non-face}) = \frac{40}{52} \]
- After drawing one non-face card, there are 51 cards left (39 non-face cards remaining). Thus, the probability for the second card is:
\[ P(\text{2nd card is non-face}) = \frac{39}{51} \]

4. **Calculate the Probability of Drawing a Face Card on the Third Turn:**
- After drawing two non-face cards, there are still 12 face cards left in the remaining 50 cards. Thus, the probability for the third card being a face card is:
\[ P(\text{3rd card is face}) = \frac{12}{50} \]

5. **Combine the Probabilities:**
- The total probability that the first two cards are non-face cards and the third card is a face card is the product of the individual probabilities:
\[ P(\text{first face card on 3rd turn}) = P(\text{1st non-face}) \times P(\text{2nd non-face}) \times P(\text{3rd face}) \]
- Substituting the values:
\[ P = \frac{40}{52} \times \frac{39}{51} \times \frac{12}{50} \]

6. **Simplify the Expression:**
- First, simplify each fraction:
\[ P = \frac{40 \times 39 \times 12}{52 \times 51 \times 50} \]
- Now calculate:
- \( 40 \) and \( 52 \) can be simplified to \( \frac{10}{13} \).
- \( 12 \) and \( 50 \) can be simplified to \( \frac{6}{25} \).
- Therefore, the expression becomes:
\[ P = \frac{10 \times 39 \times 6}{13 \times 51 \times 25} \]

7. **Calculate the Final Probability:**
- Now calculate the numerator and denominator:
- Numerator: \( 10 \times 39 \times 6 = 2340 \)
- Denominator: \( 13 \times 51 \times 25 = 16575 \)
- Thus, the probability is:
\[ P = \frac{2340}{16575} \]
- Simplifying this fraction gives:
\[ P = \frac{12}{85} \]

### Final Answer:
The probability that a face card will appear for the first time on the third turn is \( \frac{12}{85} \).