Problem 9 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Two cards are drawn from a well-shuffled pack of cards. Find the probability that both of them are aces. [MP-95, 2000]

Step-by-Step Solution

Answer

The probability that both cards drawn are aces is \( \frac{1}{221} \).

Step 1: Understand the problem

We need to determine the probability that when drawing two cards from a standard deck of 52, both are aces. The deck contains 4 aces.

Step 2: Determine the total number of ways to draw two cards

The total number of ways to draw two cards from a deck of 52 cards is given by the combination formula \( \binom{52}{2} \). Calculate this as follows: \[\binom{52}{2} = \frac{52 \times 51}{2 \times 1} = 1326.\]

Step 3: Determine the number of favorable outcomes

We need to find the number of ways to draw 2 aces from the 4 available aces. Use the combination formula \( \binom{4}{2} \) to find this:\[\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6.\]

Step 4: Calculate the probability

The probability of both cards being aces is the number of favorable outcomes divided by the total number of possible outcomes. Thus, the probability is:\[\frac{6}{1326} \approx 0.00452.\]

Step 5: Simplify the probability

Simplify the fraction \( \frac{6}{1326} \) by finding the GCD of 6 and 1326, which is 6, resulting in:\[\frac{6}{1326} = \frac{1}{221}.\]

Key Concepts

Combination Formula

The combination formula is essential in probability when determining how many different ways we can choose items from a larger set. In deck of cards scenarios, it's often used to calculate probabilities involving selections of multiple cards. This formula is denoted by \( \binom{n}{r} \), which represents the number of ways to choose \( r \) items from a set of \( n \) items without regard to order.

The formula is given by:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).

Here, \( n! \) denotes the factorial of \( n \), which is the product of all positive integers up to \( n \). The factorial concept helps us manage the orderly distribution of our choices as we calculate combinations. Understanding and applying the combination formula is crucial for solving probability problems, like the probability of drawing two aces from a deck.

Card Games

Card games are not only a popular pastime but also a fantastic way to illustrate probability concepts. In a typical card game, understanding the composition and potential draws from a deck can greatly enhance a player's strategy.

• Playing card games often involves basic probability calculations, whether considering the odds of drawing a particular card or calculating potential hands.
• Many card games require players to mentally compute odds as they play, making them practical applications of combinatorial math.

These computations usually hinge on the concepts of combinations and permutations, especially in games like poker or bridge. Studying probability through card games can thus provide both educational and strategic benefits.

Aces

Aces hold a special place in a deck of cards and probability calculations due to their unique status in most card games. Their dual role—sometimes highest, sometimes lowest—makes them fascinating subjects of probability problems.

• There are always 4 aces in a standard 52-card deck, each belonging to one of the suits: hearts, diamonds, clubs, or spades.
• When calculating the likelihood of drawing aces, configurations are essential: for example, how many ways we can draw two aces from the set of four?

This unique position often impacts the strategy in various card games, where aces might be exceptionally valuable, and likewise, they make for interesting computation challenges in probability exercises.

Deck of Cards

A standard deck of cards is a familiar tool used to teach probability because of its fixed composition and diverse applications in problems. Here's what a typical deck consists of:

• 52 cards in total.
• 4 suits: hearts, diamonds, clubs, and spades, each containing 13 cards.
• Ranks for each suit range from ace to king.

Understanding the composition of the deck is fundamental for probability problems. It helps in accurately counting the number of favorable outcomes over total possible outcomes. Since the deck is well-defined, it provides a consistent framework to calculate probabilities, such as what we've seen with the probability of drawing two aces.