Question

Five cards are dealt from a shuffled deck. What is the probability that they are all of the same suit? That they are all diamond? 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

P(all same suit) = \(\frac{4 \times \binom{13}{5}}{\binom{52}{5}}\), P(all diamonds) = \(\frac{\binom{13}{5}}{\binom{52}{5}}\), P(all face cards) = \(\frac{\binom{12}{5}}{\binom{52}{5}}\), P(sequence same suit) = \(\frac{4 \times 10}{\binom{52}{5}}\)

Step-by-step solution

Calculate the total number of ways to deal 5 cards from a deck

The total number of ways to choose 5 cards from a deck of 52 is given by the combination formula: \[ \binom{52}{5} \text{ ways} \]

Probability that all cards are of the same suit

There are 4 suits and each suit has 13 cards. The number of ways to choose 5 cards from 13 cards of the same suit is given by: \[ \binom{13}{5} \text{ for one suit} \] Since there are 4 suits, multiply by 4: \[ 4 \times \binom{13}{5} \] The probability is then: \[ P(\text{same suit}) = \frac{4 \times \binom{13}{5}}{\binom{52}{5}} \]

Probability that all cards are diamonds

Choosing 5 diamond cards from 13 diamonds: \[ \binom{13}{5} \] The probability is: \[ P(\text{all diamonds}) = \frac{\binom{13}{5}}{\binom{52}{5}} \]

Probability that all cards are face cards

There are 3 face cards (Jack, Queen, King) in each suit, giving a total of 12 face cards. The number of ways to choose 5 cards from these 12 cards is: \[ \binom{12}{5} \] The probability is: \[ P(\text{all face cards}) = \frac{\binom{12}{5}}{\binom{52}{5}} \]

Probability that the five cards are a sequence in the same suit

For a sequence in the same suit, there are 10 possible sequences (A-5, 2-6, ..., 10-K). Each sequence can be in any of the 4 suits. The number of ways to form a sequence in one suit is: \[ 10 \text{ sequences per suit} \] Since there are 4 suits, multiply by 4: \[ 4 \times 10 \] The probability is: \[ P(\text{sequence in the same suit}) = \frac{4 \times 10}{\binom{52}{5}} \]

Key concepts

combinatorial probability

Combinatorial probability combines the principles of combinatorics and probability. Combinatorics involves counting the number of ways an event can occur, while probability measures the likelihood of an event. When dealing with card probabilities, we often use combinations because the order of cards doesn’t matter.
To calculate the probability of a specific event:

• Count the number of favorable outcomes.
• Count the total number of possible outcomes.
• Divide the number of favorable outcomes by the total number of possible outcomes.

A common tool in combinatorial probability is the combination formula: \( \binom{n}{k} \), where \( n \) is the total number of items, and \( k \) is the number of items chosen. The formula is: \[ \binom{n}{k} = \frac{n!}{k!(n - k)!} \]

deck of cards

A standard deck of cards has 52 cards divided into 4 suits: hearts, diamonds, clubs, and spades. Each suit has 13 ranks: 2 through 10, Jack, Queen, King, and Ace. Understanding this structure helps in calculating probabilities related to card games.
For example, if we want to find the probability of drawing a card from a specific suit, we use the ratio of the number of cards in the suit to the total number of cards. Since each suit has 13 cards, the probability of drawing a single card from any one suit is \( \frac{13}{52} = \frac{1}{4} \).

binomial coefficients

Binomial coefficients, often written as \( \binom{n}{k} \), represent the number of ways to choose \( k \) items from \( n \) items without regard to the order. They are crucial in calculating card dealing probabilities. For instance, the total number of ways to deal 5 cards out of a 52-card deck is calculated using binomial coefficients: \[ \binom{52}{5} = \frac{52!}{5!(52 - 5)!} = 2,598,960 \]
Using binomial coefficients simplifies complex problems by reducing them into manageable computations.

probability calculations

Probability calculations involve determining the likelihood of various outcomes. For card dealing problems, it's essential to count the favourable outcomes for the event and divide it by the total number of possible outcomes.
For example, if calculating the probability of drawing five cards all of the same suit from a deck, we use:

• Number of ways to draw 5 cards from 1 suit: \( \binom{13}{5} \)
• Multiply by 4, since there are 4 suits.
• Divide by the total number of ways to draw 5 cards from 52 cards: \[ P(\text{same suit}) = \frac{4 \times \binom{13}{5}}{\binom{52}{5}} \]

Each probability calculation follows a similar structure, tailored to the specifics of the event being considered.