In the game of euchre, the deck consists of the 9, 10, jack, queen, king and ace of each suit. Players are dealt a five card hand.
What is the probability that a player is dealt 3 hearts?
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Well, since there are only 5 cards in a hand and we want to know the probability of getting 3 hearts, we can first calculate the total number of possible hands:
The total number of ways to choose 5 cards from a deck of 24 (6 hearts and 18 non-heart cards) is calculated by using combinations, which is denoted by "C" and calculated as:
C(24, 5) = 24! / (5! * (24-5)!) = 42,504
Now, we need to determine the number of favorable outcomes, which in this case is being dealt 3 hearts.
To calculate this, we need to consider the number of ways to choose 3 hearts from 6 and 2 non-heart cards from 18:
C(6, 3) * C(18, 2) = (6! / (3! * (6-3)!) ) * (18! / (2! * (18-2)!) )
= (20 * 153) = 3060
Therefore, the probability of being dealt 3 hearts in a 5-card hand from a euchre deck is:
P(3 hearts) = favorable outcomes / total outcomes
= 3060 / 42,504
≈ 0.072 or 7.2%
So, you've got around a 7.2% chance of getting 3 hearts in your euchre hand. Good luck! And if you're not a fan of hearts, maybe try calling another suit.
To calculate the probability of a player being dealt 3 hearts in a game of euchre, we need to first calculate the total number of possible hands and then determine the number of favorable outcomes.
Step 1: Calculate the total number of possible hands:
The total possible hands can be calculated using combinations. Since there are 24 cards in the deck (6 cards of each suit), we have 24C5 possible combinations for a 5-card hand.
24C5 = 24! / (5! * (24-5)!) = 42,504 possible hands.
Step 2: Determine the number of favorable outcomes:
To have 3 hearts in a 5-card hand, we need to choose 3 out of the 6 available hearts and 2 out of the remaining 18 cards in the deck.
Number of favorable outcomes = 6C3 * 18C2
= (6! / (3! * (6-3)!)) * (18! / (2! * (18-2)!))
= (6! / (3! * 3!)) * (18! / (2! * 16!))
= (6 * 5 * 4 / (3 * 2 * 1)) * (18 * 17 / (2 * 1))
= 20 * 153 = 3,060 favorable outcomes.
Step 3: Calculate the probability:
Now we can calculate the probability by dividing the number of favorable outcomes by the total number of possible hands.
Probability = Number of favorable outcomes / Total number of possible hands
= 3,060 / 42,504
= 0.072 or 7.2%
Therefore, the probability of a player being dealt 3 hearts in a game of euchre is approximately 7.2%.
To calculate the probability of being dealt 3 hearts in a game of euchre, we first need to determine the number of possible hands and the number of hands that have 3 hearts.
The total number of possible hands can be calculated using combination theory, as we need to choose 5 cards from a deck of 24 cards. The formula for combination is given by: C(n, r) = n! / (r! * (n - r)!)
In this case, we have 24 cards (the deck), and we want to choose 5 cards (the hand). So the number of possible hands is C(24, 5) = 24! / (5! * (24 - 5)!)
Next, we need to calculate the number of hands with 3 hearts. To do this, we first determine the number of ways to choose 3 hearts from the available 9 hearts in the deck. The formula for combination can be used here as well: C(n, r) = n! / (r! * (n - r)!)
In this case, we have 9 hearts in the deck, and we want to choose 3 hearts. So the number of ways to choose 3 hearts is C(9, 3) = 9! / (3! * (9 - 3)!)
After selecting the 3 hearts, we need to choose the remaining 2 cards from the remaining 15 cards in the deck (excluding the 9 hearts we already selected). This can be calculated using combination as well: C(n, r) = n! / (r! * (n - r)!)
In this case, we have 15 remaining cards, and we want to choose 2 cards. So the number of ways to choose 2 cards is C(15, 2) = 15! / (2! * (15 - 2)!)
Finally, to calculate the number of hands with 3 hearts, we multiply the number of ways to choose 3 hearts by the number of ways to choose the remaining 2 cards: C(9, 3) * C(15, 2)
The probability of being dealt 3 hearts can then be calculated by dividing the number of hands with 3 hearts by the total number of possible hands:
(C(9, 3) * C(15, 2)) / C(24, 5) = (9! / (3! * (9 - 3)!)) * (15! / (2! * (15 - 2)!)) / (24! / (5! * (24 - 5)!))
Calculating this expression will give us the probability that a player is dealt 3 hearts in the game of euchre.