Chapter 16: Problem 4

Five cards are dealt from a shuffled deck. What is the probability that they are all of the same suit? That they are all diamonds? That they are all face cards? That the five cards are a sequence in the same suit (for example, \(3,4,5,6,7\) of hearts)?

Short Answer

Expert verified

Same suit: 0.00198; All diamonds: 0.000495; All face cards: 0.000305; Sequence: 0.0000154.

Step by step solution

- Understand the Total Number of Possible Hands

First, calculate the total number of ways to deal five cards from a deck of 52 cards. This is given by the combination formula \(\binom{52}{5}\). Compute this value: \(\binom{52}{5} = \frac{52!}{5!(52-5)!} = 2,598,960\).

- Probability of All Cards Being of the Same Suit

There are 4 suits, and for each suit, there are \(\binom{13}{5}\) ways to pick 5 cards. Calculate \(\binom{13}{5} = \frac{13!}{5!(13-5)!} = 1,287\). The total number of favorable ways for any suit is \(4 \times 1,287 = 5,148\). Thus, the probability is given by \(\frac{5,148}{2,598,960} \approx 0.00198 \).

- Probability of All Cards Being Diamonds

For the case of all cards being diamonds: there are \(\binom{13}{5} = 1,287\) ways to pick 5 cards from the 13 diamonds. The probability is \(\frac{1,287}{2,598,960} \approx 0.000495 \).

- Probability of All Cards Being Face Cards

There are 12 face cards in total (Jack, Queen, King of each suit), and we need to choose 5 out of these 12: \(\binom{12}{5} = 792 \). The probability is \(\frac{792}{2,598,960} \approx 0.000305 \).

- Probability of the Five Cards Forming a Sequence in the Same Suit

For each suit, there are sequences of 5 cards (e.g., \(3,4,5,6,7\)). The possible sequences for each suit can start from Ace to 10 (Ace, 2, 3, 4, 5 to 10, Jack, Queen, King), so there are 10 sequences per suit. With 4 suits, this gives us \(4 \times 10 = 40\) sequences. The probability is \(\frac{40}{2,598,960} \approx 0.0000154 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinations in Card Games

Combinations are a way of selecting items from a larger pool where the order does not matter. In card games, we often use combinations to determine the number of ways to form a specific hand.
For example, the number of ways to pick 5 cards from a deck of 52 cards is given by the combination formula: \(\binom{52}{5}\).
This is calculated as follows:
\[\binom{52}{5} = \frac{52!}{5!(52-5)!}\]
Where \(n!\) represents the factorial of \(n\) (i.e., the product of all positive integers up to n). For our case, \(\binom{52}{5} = 2,598,960\).
Combinations are a fundamental tool in calculating probabilities in card games.

Probability of Same Suit

Finding the probability that all five cards dealt are of the same suit involves a few steps. First, we calculate how many ways we can pick 5 cards from any given suit of 13 cards, which is \(\binom{13}{5}\).
This is computed as:
\[\binom{13}{5} = \frac{13!}{5!(13-5)!} = 1,287\]
Since there are 4 suits, the total number of favorable outcomes is \(\binom{13}{5} \times 4 = 5,148\).
Therefore, the probability of being dealt 5 cards all of the same suit is:
\[\frac{5,148}{2,598,960} \text{ or approximately } 0.00198\]
This small probability indicates it's quite rare to be dealt five cards all of the same suit.

Probability of Face Cards

Face cards in a deck include Jacks, Queens, and Kings. Each suit has 3 face cards, making a total of 12 face cards in the whole deck.
To find the probability of being dealt 5 face cards, we first calculate the number of ways to pick 5 from these 12 face cards:
\[\binom{12}{5} = \frac{12!}{5!(12-5)!} = 792\]
The number of ways to deal any 5 cards from the full deck is still \(2,598,960\). Therefore, the probability is given by:
\[\frac{792}{2,598,960} \text{ or approximately } 0.000305\]
So, it's quite rare to be dealt 5 face cards in a single hand.

Probability of Sequences

A sequence means the cards follow one another in numerical order, and we're specifically considering sequences within the same suit.
For any suit, sequences of 5 cards can start at any rank from Ace to 10 (e.g., A, 2, 3, 4, 5 through 10, J, Q, K), giving us 10 possible sequences per suit. With 4 suits, there are a total of:
\4 \times 10 = 40\ sequences.
Thus, the probability of the 5 cards being a sequence in the same suit is:
\[\frac{40}{2,598,960} \text{ or approximately } 0.0000154\]
Again, this shows that such an event is exceedingly rare.