Problem 55 - A First Course in Probability

Question

Compute the probability that a hand of 13 cards contains (a) the ace and king of at least one suit; (b) all 4 of at least 1 of the 13 denominations.

Step-by-Step Solution

Answer

(a) The probability of having the Ace and King of at least one suit in a 13-card hand is \[ P(\text{Ace and King of at least one suit}) = \frac{\binom{52}{13} - \binom{48}{13}}{\binom{52}{13}} \] (b) The probability of having all 4 cards of at least one of the 13 denominations in a 13-card hand is \[ P(\text{all 4 cards of at least one denomination}) = \frac{\binom{13}{1} \cdot \binom{48}{9}}{\binom{52}{13}} \]

Step 1: Calculate total number of possible hands

First, we have to calculate the total number of 13-card hands we can get from a standard 52-card deck. This is given by the combination formula: \[ \text{total number of hands} = \binom{52}{13} \]

Step 2: (a) Calculate favorable hands for Ace and King of at least one suit

1. Calculate the number of hands that have the Ace and King of no suit: \[ \text{hands without Ace and King} = \binom{48}{13} \] 2. Subtract the number of hands without Ace and King from the total number of hands: \[ \text{favorable hands} = \binom{52}{13} - \binom{48}{13} \] 3. Calculate the probability of getting at least one Ace and King of the same suit by dividing the number of favorable hands by the total number of hands: \[ P(\text{Ace and King of at least one suit}) = \frac{\binom{52}{13} - \binom{48}{13}}{\binom{52}{13}} \]

Step 3: (b) Calculate favorable hands for all 4 cards of at least one denomination

1. Calculate the number of ways we can choose 1 denomination out of 13 to have all 4 cards: \[ \text{choose 1 denomination} = \binom{13}{1} \] 2. Calculate the number of ways to choose the remaining 9 cards from the remaining 48 (since we excluded 4 cards from the chosen denomination): \[ \text{choose 9 other cards} = \binom{48}{9} \] 3. Multiply the number of ways to choose 1 denomination and the number of ways to choose the remaining 9 cards to find the number of favorable hands: \[ \text{favorable hands} = \binom{13}{1} \cdot \binom{48}{9} \] 4. Calculate the probability of having all 4 cards of at least one denomination by dividing the number of favorable hands by the total number of hands: \[ P(\text{all 4 cards of at least one denomination}) = \frac{\binom{13}{1} \cdot \binom{48}{9}}{\binom{52}{13}} \] Now that we have the probabilities for (a) and (b), we can provide the final answers: (a) The probability of having the Ace and King of at least one suit in a 13-card hand is \[ P(\text{Ace and King of at least one suit}) = \frac{\binom{52}{13} - \binom{48}{13}}{\binom{52}{13}} \] (b) The probability of having all 4 cards of at least one of the 13 denominations in a 13-card hand is \[ P(\text{all 4 cards of at least one denomination}) = \frac{\binom{13}{1} \cdot \binom{48}{9}}{\binom{52}{13}} \]

Key Concepts

Combinatorics

Combinatorics is a branch within mathematics that deals with counting, arranging, and combining objects. It provides tools and techniques essential for solving problems involving selections and arrangements. In cards, we often use combinatorics when figuring out the number of possible hands or different combinations of cards.

One fundamental concept in combinatorics is the "combination." This refers to selecting a group of items from a larger pool where the order does not matter. For instance, when selecting 13 cards from a deck of 52, the number of ways to do this is expressed by the formula\[ \binom{52}{13} \]. This symbol, pronounced as "52 choose 13," calculates the number of combinations of 13 items from a set of 52. The formula to calculate combinations is:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items to choose from, \( r \) is the number of items to choose, and \( ! \) denotes factorial, which is the product of all positive integers up to that number.

Card Combinations

In a card game setting, understanding card combinations is crucial. Let's delve a bit into how combinations relate to drawing cards from a deck. When calculating the probability of having specific combinations like an ace and king of at least one suit, combinatorics become handy.

For example, to determine the number of hands without any ace and king from any suit, we use:\[ \binom{48}{13} \]
This tells us the number of 13-card hands formed from the 48 cards left after removing all aces and kings. To find the number of hands containing at least one ace and king, we subtract this from the total number of hands:
\[ \binom{52}{13} - \binom{48}{13} \]
These calculations help in determining various possible and favorable card hand scenarios in card games, aiding strategic decisions.

Probability Theory

Probability theory is the mathematical framework used to evaluate the likelihood of various outcomes. It involves calculating the chances of specific events happening. For example, in card games, understanding the chances of drawing specific cards plays a vital role in strategy.

In the provided problem, we calculated probabilities for different card-hand scenarios, such as having an ace and king of at least one suit or all four cards of at least one denomination. These probabilities are computed by dividing the number of favorable outcomes by the total number of possible outcomes. For instance, to find the probability of having the ace and king of at least one suit in a 13-card hand, we use:
\[ P(\text{Ace and King of at least one suit}) = \frac{\binom{52}{13} - \binom{48}{13}}{\binom{52}{13}} \]
This formula gives us a number between 0 and 1, which represents the likelihood of the event occurring.

Understanding probability theories aids in making informed predictions on various events' outcomes, be it card games or other areas of life requiring risk calculations.