To solve this problem, let's follow these steps:
1. Identify the Total Number of Cards and Face Cards
- A standard deck has 52 cards.
- There are 12 face cards in a deck (4 Jacks, 4 Queens, and 4 Kings).
2. Determine the Number of Non-Face Cards
- Non-face cards are those that are not Jacks, Queens, or Kings. Therefore, there are [tex]\( 52 - 12 = 40 \)[/tex] non-face cards.
3. Calculate the Number of Ways to Choose 4 Face Cards out of 12
- The number of ways to choose 4 face cards from 12 face cards can be calculated using the combination formula:
[tex]\[
\binom{12}{4} = \frac{12!}{4!(12-4)!}
\][/tex]
Simplifying, we get 495 ways.
4. Calculate the Number of Ways to Choose 2 Non-Face Cards out of 40
- Similarly, the number of ways to choose 2 non-face cards from 40 non-face cards is:
[tex]\[
\binom{40}{2} = \frac{40!}{2!(40-2)!}
\][/tex]
This simplifies to 780 ways.
5. Determine the Total Number of Ways to Choose 6 Cards from 52 Cards
- The total number of ways to choose any 6 cards from a deck of 52 cards is given by:
[tex]\[
\binom{52}{6} = \frac{52!}{6!(52-6)!}
\][/tex]
This evaluates to 20,358,520 ways.
6. Calculate the Probability of Drawing a Hand with Exactly 4 Face Cards
- The probability is found by dividing the number of favorable outcomes by the total number of possible outcomes:
[tex]\[
\text{Probability} = \frac{\binom{12}{4} \times \binom{40}{2}}{\binom{52}{6}}
\][/tex]
Substituting the values, we get:
[tex]\[
\text{Probability} = \frac{495 \times 780}{20,358,520} \approx 0.018965032821639295
\][/tex]
7. Round the Probability to 4 Decimal Places
- The final probability rounded to four decimal places is:
[tex]\[
\boxed{0.0190}
\][/tex]
Thus, the probability that a 6-card poker hand contains exactly 4 face cards is 0.0190 when rounded to four decimal places.
