Problem 47 - Calculus for Biology and Medicine

Question

Thirteen cards are drawn at random without replacement from a standard deck of 52 cards. What is the probability that all are red?

Step-by-Step Solution

Answer

The probability all cards drawn are red is \( \frac{\binom{26}{13}}{\binom{52}{13}} \).

Step 1: Identify Possible Outcomes

A standard deck has 52 cards. Each time we draw a card, the total number of remaining cards decreases. The total number of ways to choose 13 cards from 52 is given by the combination formula \( \binom{n}{k} \), which is \( \binom{52}{13} \).

Step 2: Identify Favorable Outcomes

We want all 13 cards drawn to be red. There are 26 red cards (hearts and diamonds) in the deck. The number of ways to choose 13 red cards from these 26 is \( \binom{26}{13} \).

Step 3: Calculate Combinatorial Values

Calculate \( \binom{52}{13} \) using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), giving \( \binom{52}{13} = \frac{52!}{13!39!} \). Also, calculate \( \binom{26}{13} = \frac{26!}{13!13!} \).

Step 4: Compute the Probability

The probability that all 13 cards drawn are red is given by the ratio of favorable outcomes to possible outcomes: \[ P(\text{all red}) = \frac{\binom{26}{13}}{\binom{52}{13}} \]. Calculate this value to find the probability.

Key Concepts

Combinatorics

Combinatorics is a key mathematical concept that deals with counting and arrangements. It helps us figure out how many different ways things can be combined or ordered. In problems like the card-drawing scenario, we use combinators to find the total possible outcomes.

• The combination formula, represented as \( \binom{n}{k} \), is used to determine the number of ways to choose \( k \) elements from \( n \) elements without regard to order.
• In this exercise, \( \binom{52}{13} \) calculates the total possible ways to draw 13 cards from a 52-card deck.
• Similarly, \( \binom{26}{13} \) calculates the number of ways to choose 13 red cards from the 26 red cards available in the deck.

Combinatorial calculations are essential for determining the likelihood of specific outcomes in probability and statistical scenarios. Understanding how to select and arrange components using combinations is crucial for solving complex probability problems.

Probability Theory

Probability theory is the branch of mathematics that deals with calculating the likelihood of different events. It provides a framework for making predictions and understanding uncertain outcomes. In the problem of drawing cards, probability theory helps us quantify the chance of drawing all red cards.

• Probability is often expressed as a fraction that ranges from 0 (impossible event) to 1 (certain event).
• The formula for probability is \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \).
• In our card example, to find the probability all cards are red, we calculate \( P(\text{all red}) = \frac{\binom{26}{13}}{\binom{52}{13}} \).

Probability theory helps us make informed decisions based on the likelihood of events, which is crucial in fields ranging from games to scientific research and engineering.

Random Selection

Random selection involves choosing items from a set without any specific plan or pattern. It ensures that each possible outcome has an equal chance of occurring. In the context of drawing cards, random selection means each card drawn has an equal probability of being picked, as long as we don't know its value beforehand.

• In a standard deck of 52 cards, each card is equally likely to be drawn first.
• When we draw cards without replacement, the conditions change as each subsequent draw depends on the previous ones.
• This change impacts the total number of possibilities and is an important factor in probability calculations.

Understanding random selection is integral to questions involving probability, as it lays the foundation for fair, unbiased outcomes. Mastering this concept enables you to apply probability principles to real-world scenarios and theoretical exercises proficiently.