To find the probability that a 6-card poker hand contains exactly 3 face cards, we need to calculate the total number of favorable outcomes (hands with 3 face cards) divided by the total number of possible outcomes (all possible 6-card hands).
First, let's determine the total number of possible outcomes:
The number of ways to choose 6 cards out of a standard deck of 52 cards is given by the combination formula: C(52, 6).
C(52, 6) = 52! / (6!(52-6)!) = 22,957,480.
Next, let's calculate the number of favorable outcomes:
To have exactly 3 face cards in a 6-card hand, we need to choose 3 face cards from the 12 available face cards (3 out of 12) and 3 non-face cards from the remaining 40 non-face cards (3 out of 40). The order of the cards doesn't matter in this case.
The number of ways to choose 3 face cards from 12 is given by: C(12, 3) = 12! / (3!(12-3)!) = 220.
Similarly, the number of ways to choose 3 non-face cards from 40 is given by: C(40, 3) = 40! / (3!(40-3)!) = 9,880.
To find the number of favorable outcomes, we multiply these two values together:
Favorable outcomes = C(12, 3) * C(40, 3) = 220 * 9,880 = 2,173,600.
Now, we can calculate the probability:
Probability = Favorable outcomes / Total outcomes = 2,173,600 / 22,957,480 ≈ 0.0944.
Therefore, the probability that a 6-card poker hand contains exactly 3 face cards is approximately 0.0944 or 9.44%.