Special Euchre desk is a card game that uses only the Aces, Kings, Queens, Jacks, 10s, 9s, 8s, 7s of each suit. Seven-card hands are dealt to the players. What is the probability of selecting five face cards?
To calculate the probability of selecting five face cards in a seven-card hand in Special Euchre, we need to determine two things: the total number of possible hands and the number of hands that have five face cards.
Step 1: Calculate the total number of possible hands.
In Special Euchre, we only use the Aces, Kings, Queens, Jacks, 10s, 9s, 8s, and 7s of each suit. There are 32 cards in total (8 cards per suit x 4 suits). To calculate the total number of possible hands, we use the combination formula:
C(n, r) = n! / (r!(n - r)!)
where:
n is the total number of cards (32 in this case)
r is the number of cards we are selecting (7 in this case)
So, the total number of possible hands is:
C(32, 7) = 32! / (7!(32 - 7)!) = 143,925
Step 2: Calculate the number of hands that have five face cards.
In Special Euchre, there are a total of 14 face cards (4 Kings, 4 Queens, 4 Jacks, and 2 Aces). To calculate the number of hands with five face cards, we consider two scenarios:
1. Selecting 5 face cards and 2 non-face cards.
- The number of ways to select 5 out of 14 face cards is C(14, 5).
- The number of ways to select 2 non-face cards from the remaining 18 cards (32 total cards minus the 14 face cards) is C(18, 2).
- So, the number of hands with five face cards and two non-face cards is: C(14, 5) x C(18, 2).
2. Selecting 6 face cards and 1 non-face card.
- The number of ways to select 6 out of 14 face cards is C(14, 6).
- The number of ways to select 1 non-face card from the remaining 18 cards is C(18, 1).
- So, the number of hands with six face cards and one non-face card is: C(14, 6) x C(18, 1).
Add the results from the two scenarios to get the total number of hands with five face cards:
C(14, 5) x C(18, 2) + C(14, 6) x C(18, 1) = 65,100
Step 3: Calculate the probability.
The probability of selecting five face cards in a seven-card hand is the number of hands with five face cards divided by the total number of possible hands:
Probability = (Number of hands with five face cards) / (Total number of possible hands)
= 65,100 / 143,925
= 0.4521 (rounded to four decimal places)
Therefore, the probability of selecting five face cards in a seven-card hand in Special Euchre is approximately 0.4521 or 45.21%.