Problem 45 - Calculus for Biology and Medicine

Question

Four cards are drawn at random without replacement from a standard deck of 52 cards. What is the probability of at least one ace?

Step-by-Step Solution

Answer

The probability of drawing at least one ace is approximately 0.384.

Step 1: Understanding the Problem

We need to find the probability of getting at least one ace when four cards are drawn without replacement from a standard deck of 52 cards. The deck contains 4 aces out of 52 cards.

Step 2: Total Possible Outcomes

Calculate the total number of possible outcomes when drawing 4 cards from a standard deck. This can be done using combinations: \[C(52, 4) = \frac{52!}{4!(52-4)!} = 270725\]

Step 3: Calculate Probability of No Aces

Next, calculate the probability that none of the cards is an ace. First, determine the number of ways to draw 4 cards from the 48 non-ace cards. \[C(48, 4) = \frac{48!}{4!(48-4)!} = 194580\]

Step 4: Complement Rule

Using the complement rule, the probability of at least one ace is equal to 1 minus the probability of no aces. The probability of no aces is given by the number of combinations without aces divided by the total number of combinations:\[P(\text{No Aces}) = \frac{194580}{270725}\]Then, calculate:\[P(\text{At least one Ace}) = 1 - P(\text{No Aces})\]

Step 5: Calculate and Simplify

Calculate the probability:\[P(\text{At least one Ace}) = 1 - \frac{194580}{270725} = \frac{76145}{270725} \approx 0.384\]

Key Concepts

Understanding Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects. It's crucial in calculating probabilities in scenarios where different groups can be formed. Consider the problem of selecting 4 cards from a 52-card deck. This is where combinatorics comes into play.
We use combinatorial math to figure out the various ways we can select subsets of cards. The formula for combinations is given by

• \( C(n, k) = \frac{n!}{k!(n-k)!} \)

Here, \( n \) is the total number of items to choose from, \( k \) is the number of items to choose, and "!" denotes factorial, which means multiplying a series of descending natural numbers. In our exercise, we calculated

• \( C(52, 4) \), representing the total number of different ways to choose 4 cards from 52.
• \( C(48, 4) \), showing how to choose 4 cards from the 48 non-ace cards.

Using combinatorics, we quantify possible selections, essential for probability analysis.

Embracing the Complement Rule

The complement rule is a fundamental concept in probability theory. It helps us simplify calculations by focusing on the opposite of the desired event. The principle asserts that the probability of an event happening is one minus the probability of its complement event.
For instance, if we need to determine the probability of drawing at least one ace from the deck, we can first calculate the probability of the opposite - not drawing any ace. Doing this is sometimes easier and quick. Once found, we subtract this probability from 1, as shown in the step-by-step solution.

• \( P(\text{At least one Ace}) = 1 - P(\text{No Aces}) \)
• This approach means fewer computations and results in the same accurate probability.

Complement rule offers a practical, efficient way to tackle probability problems, often simplifying more complex calculations.

Card Probability Explained

Card probability deals with finding the likelihood of certain outcomes when drawing cards from a deck. Working with a standard 52-card deck, understanding the structure of the deck is fundamental. It consists of 4 suits and 13 different ranks per suit. Some decks include jokers, but they are usually omitted in probability exercises like ours.
To solve our problem, we calculated:

• The total ways of drawing 4 cards: \( C(52, 4) \).
• The configurations without any aces: \( C(48, 4) \).

The ratio of these helps us in determining no-ace probability, and thereby using the complement rule, finding the chance of at least one ace.
Probability is thus a function of the selection options from the entire deck, and by understanding these relationships, we can systematically calculate desired odds.