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 that all 13 cards are red is approximately 0.000016385.

Step 1: Understanding the Problem

We must find the probability that all 13 cards drawn from a deck of 52 are red. In a standard deck, half the cards, 26, are red. We are drawing without replacement, meaning once a card is drawn, it is not put back into the deck.

Step 2: Total Number of Ways to Choose 13 Cards

Calculate the total number of ways to draw 13 cards from 52. This is given by the binomial coefficient, which is computed as: \[ \binom{52}{13} \] This represents the total number of combinations of 13 cards that can be picked from a set of 52.

Step 3: Number of Ways to Choose 13 Red Cards

We need to compute the number of ways to choose 13 red cards from the 26 red cards available. This is another binomial coefficient:\[ \binom{26}{13} \] This represents the total number of combinations of 13 cards that can be selected from the 26 red cards.

Step 4: Calculating the Probability

The probability that all 13 cards drawn are red is calculated by dividing the number of favorable outcomes (choosing 13 red cards) by the total number of outcomes (choosing any 13 cards). Thus, the probability is:\[ P(\text{All 13 are red}) = \frac{\binom{26}{13}}{\binom{52}{13}} \]

Step 5: Solving the Combinations

Calculate \(\binom{26}{13}\) and \(\binom{52}{13}\) using a calculator or software:\[ \binom{26}{13} = 10,400,600 \]\[ \binom{52}{13} = 635,013,559,600 \]

Step 6: Compute Probability Value

Substitute the values obtained into the probability formula:\[ P(\text{All 13 are red}) = \frac{10,400,600}{635,013,559,600} \approx 0.000016385 \]

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that focuses on counting, arranging, and finding patterns. It's a powerful tool for solving problems that involve selection and arrangement. In this context, you want to know how many ways things can occur. In the problem of drawing cards, combinatorics helps determine how many different sets of 13 cards we can draw. It does this by calculating combinations of elements. Combinations are selections of items where the order does not matter. This is essential when you select cards from a deck, as the sequence doesn't change the selection. The number of combinations is what the binomial coefficient calculates, allowing you to determine total possibilities in situations where order is irrelevant. Thus, using combinatorics, you can figure out the total possible outcomes when drawing cards, as well as the desired outcomes, leading us to the calculation of probability.

Binomial Coefficient

The binomial coefficient is a symbol used in combinatorics to represent the number of ways you can choose a subset of items from a larger set. It is often pronounced as "n choose k" and written as \( \binom{n}{k} \). Here, \( n \) is the total number of items to choose from, and \( k \) is the number of items to be chosen.The formula for the binomial coefficient is:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]Where \( n! \) (n factorial) is the product of all positive integers up to \( n \). For the card problem, you use \( \binom{52}{13} \) to find the total possible ways to select 13 cards from 52. Similarly, \( \binom{26}{13} \) calculates the ways to choose 13 red cards from 26 red cards.By plugging into these equations, you obtain numbers that reflect all possible selections for the specific case of 13 cards drawn from a standard deck.

Probability without Replacement

Probability without replacement refers to scenarios where once an item is chosen from a set, it is not returned to the set before the next selection. This is crucial in card problems like drawing from a deck; once a card is drawn, it cannot be drawn again unless the drawing starts anew.In our deck problem, this changes the way probability is calculated:

• Each choice affects the next, decreasing the total number of cards available.
• This impacts the probability of drawing a particular card as subsequent draws happen from a reduced pool.

When you calculate probability without replacement, you depend on combinatorial logic to determine favorable outcomes over all possible outcomes. Here, this manifests as dividing the number of ways to choose all red cards (desired outcome) by all possible choices of cards (total outcome), giving:\[ P(\text{All 13 are red}) = \frac{\binom{26}{13}}{\binom{52}{13}} \]Understanding how replacement or lack thereof affects calculations is critical in probability to reflect true odds.