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# Euchre: What are the Odds?

Updated: May 22, 2023  In Euchre, the deck consists of 24 cards, comprising four suits – spades, hearts, diamonds, and clubs. Each suit contains six cards: 9, 10, Jack, Queen, King, and Ace. While some card combinations may seem elusive and near impossible to obtain, understanding the probabilities involved can add an extra layer of intrigue to the game.

In this article, we will explore explore the chances of rare card combinations, offering insights for the most curious players.

## 1. Getting All Four Jacks

To calculate the probability of a player receiving all four Jacks in a 24-card deck Euchre game, we can use the concept of combinations.

First, let’s determine the total number of possible 5-card hands that a player can receive from a 24-card deck. We’ll use the notation “C(n, k)” to represent the combination of selecting k items from a set of n items. The formula for combinations is:

C(n, k) = n! / (k! * (n – k)!)

In this case, n = 24 (total number of cards in the deck) and k = 5 (number of cards in each hand).

Total possible hands = C(24, 5) = 24! / (5! * (24 – 5)!) = 42,504

Now, let’s calculate the number of favorable outcomes, which is the number of hands that contain all four Jacks.

Number of favorable outcomes = C(4, 4) * C(20, 1) = 1 * 20 = 20

The combination C(4, 4) accounts for the selection of all four Jacks, and C(20, 1) represents the selection of one card from the remaining 20 cards in the deck.

Finally, we can calculate the probability by dividing the number of favorable outcomes by the total possible hands:

Probability = Number of favorable outcomes / Total possible hands = 20 / 42,504 ≈ 0.000471

Therefore, the probability of a player receiving all four Jacks in a 24-card deck Euchre game is approximately 0.000471, or about 0.0471%.

## 2. Getting Five Cards of the Same Suit

When aiming for a hand composed entirely of cards from a single suit, the probability varies based on the number of cards of that suit in the deck. For instance, if the deck contains six cards of a specific suit, such as spades, the probability of receiving all five spade cards is approximately 0.000141 or about 0.0141%.

Probability = Number of favorable outcomes / Total possible hands = 6 / 42,504 ≈ 0.000141

## 3. Getting No Face Cards

To calculate the probability of receiving no face cards (Jacks, Kings, and Queens), we consider the number of favorable outcomes and the total possible hands. With 12 face cards in the deck, the probability of avoiding them entirely is approximately 0.0186 or about 1.86%.

Number of favorable outcomes = C(12, 0) * C(12, 5) = 1 * 792 = 792

The combination C(12, 0) represents the selection of zero face cards, and C(12, 5) represents the selection of 5 non-face cards from the remaining 12 cards in the deck.

Probability = Number of favorable outcomes / Total possible hands = 792 / 42,504 ≈ 0.0186

## 4. Getting Only 9s and 10s

The probability of receiving a hand consisting only of 9s and 10s in a 24-card deck Euchre game depends on the number of 9s and 10s in the deck. With a total of eight 9s and 10s available, the number of favorable outcomes is 56. This gives us a probability of approximately 0.00132 or about 0.132%.

## 5. Getting the Joker Card

If we add a single Joker card to the 24-card deck, the total number of cards becomes 25. To calculate the probability of a player receiving the Joker card in their hand, we’ll consider the total possible hands and the number of favorable outcomes.

Probability = Number of favorable outcomes / Total possible hands = 10,626 / 53,130 ≈ 0.200.

Therefore, the probability of a player receiving the Joker card in their hand in a 25-card deck Euchre game is approximately 0.200, or 20%.

## 6. Going Alone and Winning Three Hands

The odds of winning three hands when going alone in Euchre depend on the winning probability in a single hand. Considering the number of combinations, which is 10 for winning 3 out of 5 hands, multiplying this by the winning probability will give the specific odds for the given scenario.
Understanding the odds in a game of Euchre can greatly enhance a player’s decision-making process and strategic gameplay. From aiming for specific card combinations to evaluating the likelihood of winning in various scenarios, having a grasp of these probabilities can give players a competitive edge.
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