Question

When two cards are drawn from a well-shuffled pack of cards, what is the probability that both will be aces? (1) \(\frac{1}{221}\) (2) \(\frac{2}{221}\) (3) \(\frac{1}{13}\) (4) \(\frac{1}{231}\)

Short answer

Answer: \(\frac{1}{221}\)

Step-by-step solution

Determine the probability of drawing the first ace

There are 4 aces in a deck of 52 cards. When we draw the first card, the probability of it being an ace is the number of aces (4) divided by the total number of cards (52): \(\frac{4}{52} = \frac{1}{13}\).

Determine the probability of drawing the second ace

After drawing the first ace, there are now only 3 aces left in the deck, and there are 51 cards remaining. The probability of drawing a second ace is the number of remaining aces (3) divided by the total number of remaining cards (51): \(\frac{3}{51}\).

Calculate the probability of both events happening

To find the probability of both events (drawing both aces), multiply the probabilities from Step 1 and Step 2: \(\frac{1}{13} \times \frac{3}{51} = \frac{3}{663}\).

Simplify the probability

The probability can be simplified by finding the greatest common divisor and dividing both numerator and denominator by it: \(\frac{3}{663} = \frac{1}{221}\).

Identify the correct answer

The probability of drawing two aces from a well-shuffled deck is \(\frac{1}{221}\). Therefore, the correct answer is option (1).

Key concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and grouping objects. When dealing with probabilities, it's essential to determine how many ways a certain event can occur compared to all possible outcomes. In our exercise, we're interested in the probability of drawing two aces in a row from a deck of cards. To do this, we use combinatorial methods to calculate each step efficiently. We first find the number of favorable outcomes for each event (drawing an ace) and then determine how these events interact probabilistically.

Deck of Cards

A standard deck has 52 cards, encompassing 4 suits (hearts, diamonds, clubs, spades), each with 13 ranks (Ace through King). Understanding the composition of a deck is crucial for solving probability problems. In this context, we're interested in aces, of which there are exactly 4 in a deck: the Ace of Hearts, Diamonds, Clubs, and Spades. Knowing these basics allows us to calculate the likelihood of drawing specific cards during a probability problem. For instance, the initial chance of drawing one ace out of the deck is dependent on its proportion among all cards.

Simplifying Fractions

Simplifying fractions is about reducing them to their simplest form. This is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD). For example, the fraction \( \frac{3}{663} \) can be simplified by dividing both 3 and 663 by 3, resulting in \( \frac{1}{221} \). Simplification is important for clarity and often required as a final step in probability problems, ensuring the solution is presented in its most understandable form.

Step-by-Step Solution

Following a step-by-step approach helps in understanding intricate mathematical processes. Breaking down the exercise into sequential steps, like determining individual probabilities for consecutive card draws, clarifies the overall calculation. Initially, the probability of drawing the first ace is calculated. Then, we consider the changed conditions for the second draw, with one ace and one card removed from the deck. Each step builds on the last, illustrating how individual actions affect the total probability. Finally, simplifying the resulting fraction reveals the final answer clearly, promoting a comprehensive understanding of the problem.