Problem 50 - Calculus for Biology and Medicine

Question

Five cards are drawn at random without replacement from a standard deck of 52 cards. What is the probability of three of a kind and a pair (for instance, \(\mathrm{Q} \mathrm{Q} \mathrm{Q} 33) ?\) (This is called a full house in poker.)

Step-by-Step Solution

Answer

The probability of getting a full house is approximately 0.144%.

Step 1: Understanding the problem

We need to calculate the probability of drawing five cards from a deck that form a 'full house,' which consists of three cards of one rank and two cards of another rank.

Step 2: Determine the total number of possible hands

The total number of ways to draw 5 cards from a deck of 52 cards is given by the combination formula \(\binom{n}{k}\): \[\binom{52}{5}\]Calculating this, we have:\[\binom{52}{5} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2,598,960\]

Step 3: Determine the number of ways to get three of a kind

To form a three of a kind, select one rank from the available 13. \(\binom{13}{1} = 13\).For each rank, select 3 out of 4 cards with that rank: \(\binom{4}{3} = 4\).So, there are \(13 \times 4\) ways to choose three cards of the same rank.

Step 4: Determine the number of ways to get a pair

After selecting the rank for three of a kind, 12 ranks remain for the pair.Choose one of these ranks: \(\binom{12}{1} = 12\).For that rank, choose 2 cards from the 4 available of that rank: \(\binom{4}{2} = 6\).So, there are \(12 \times 6\) ways to choose the pair.

Step 5: Calculate total number of favorable outcomes

Multiply the number of ways to select three of a kind by the number of ways to select a pair:\((13 \times 4) \times (12 \times 6) = 13 \times 4 \times 12 \times 6 = 3,744\) ways to form a 'full house.'

Step 6: Calculate the probability

The probability of drawing a 'full house' is given by the favorable outcomes divided by the total possible hands.\[P(\text{full house}) = \frac{3,744}{2,598,960} \approx 0.00144\]

Step 7: Finalize the probability value

Therefore, the probability of drawing a full house is approximately 0.144%.

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics focused on counting, arranging, and combining objects. It's like a mathematical toolkit that helps us solve problems related to selection and organization.
One significant application of combinatorics is understanding how to form different combinations from a set of items. With combinatorics, you can figure out how many ways you can pick a subset of items from a larger group, which is especially handy in probability and statistics.

• **Combinations**: In probabilistic terms, combinations are a way to determine how many ways you can choose a subset of items without considering the order. For example, determining the number of five-card hands possible from a deck of cards involves combinations.
• **Formulas**: The formula for combinations is given by \( \binom{n}{k} \), which is read as "n choose k." It calculates the number of ways to choose k items from n without regard to order: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

Combinatorics becomes essential when calculating probabilities, as it helps determine the total number of possible favorable outcomes.

Card Games

Card games provide an excellent playground for applying probability concepts. They involve a variety of combinatorial strategies because they rely on random draws from a deck, where understanding the odds is key.

• **Types of Games**: Games like poker, bridge, and blackjack depend heavily on probability and strategic planning. Players calculate odds to make decisions, like whether to draw another card or keep their hand.
• **Hand Ranking**: Different card hands have various values. Understanding the probabilities of different hands helps players assess risks and opportunities during the game.

The skills gained from playing card games extend beyond the table, offering insights into decision-making and risk assessment in real-life scenarios. They require and develop an understanding of basic probability principles.

Full House

In poker, a 'full house' is a powerful and well-valued hand. It consists of three cards of one rank and two cards of another rank. Let’s dive into how to calculate the probability of drawing this hand.

• **Formation**: To create a full house, first choose one rank for the three of a kind. Then, select another distinct rank for the pair. Use combination formulas to determine how many ways you can create each part of the full house.
• **Calculations**: The process involves two main steps:
  1. Choose the rank for three of a kind: \( \binom{13}{1} \), and choose 3 cards from 4 of that rank: \( \binom{4}{3} \). 2. Select a different rank for the pair: \( \binom{12}{1} \), and choose 2 cards from 4 of that rank: \( \binom{4}{2} \).

Ultimately, by combining these steps, you can find out the total number of ways to form a full house, which is crucial when calculating its probability of occurrence.

Deck of Cards

A standard deck of cards is a structured set containing 52 individual cards, each belonging to one of four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 ranks ranging from Ace through King.

• **Structure**: Understanding the structure of a deck is key when solving probability problems, as it provides the context for calculations and strategy in card games.
• **Possibilities**: Each card draw represents a different combination of suits and ranks, influencing the odds of drawing specific hands like flushes, straights, or full houses.
• **Replacement and Non-Replacement**: Calculations often consider whether cards are drawn with or without replacement. For example, drawing without replacement means that once a card is drawn, it can't be drawn again, affecting the overall odds with each draw.

Understanding these fundamentals helps in assessing probabilities and making informed decisions in card games and other combinatorial problems.