Problem 50 - Algebra and Trigonometry

Question

A card is drawn from a deck and replaced, and then a second card is drawn. (a) What is the probability that both cards are aces? (b) What is the probability that the first is an ace and the second a spade?

Step-by-Step Solution

Answer

(a) \(\frac{1}{169}\); (b) \(\frac{1}{52}\).

Step 1: Determine Total Possible Outcomes

A standard deck of cards has 52 cards. Since the first card is drawn and replaced, the total number of possible outcomes for each draw is 52. Therefore, for two draws, there are \(52 \times 52 = 2704\) possible outcomes.

Step 2: Calculate Probability of First Event - Two Aces

There are 4 aces in a deck. The probability of drawing an ace on the first draw is \(\frac{4}{52}\). Since the card is replaced, the probability of drawing an ace again is the same for the second draw, \(\frac{4}{52}\). Thus, the probability of both events occurring is \(\left(\frac{4}{52}\right) \times \left(\frac{4}{52}\right)\).

Step 3: Simplify Probability Calculation for Two Aces

The probability of drawing two aces is \(\frac{4}{52} \times \frac{4}{52} = \frac{16}{2704}\). Simplifying this fraction gives us \(\frac{1}{169}\).

Step 4: Calculate Probability of First Event - Ace then Spade

The probability of drawing an ace first is \(\frac{4}{52}\) as previously calculated. A deck has 13 spades, so the probability of drawing a spade on the second draw is \(\frac{13}{52}\). Thus, the probability of drawing an ace followed by a spade is \(\frac{4}{52} \times \frac{13}{52}\).

Step 5: Simplify Probability Calculation for Ace then Spade

The probability of drawing an ace then a spade is \(\frac{4}{52} \times \frac{13}{52} = \frac{52}{2704}\). Simplifying this fraction gives us \(\frac{1}{52}\).

Key Concepts

Independent Events

In probability, independent events are situations in which the outcome of one event does not affect the outcome of another. For instance, when drawing a card from a deck and then replacing it, the next card drawn is unaffected by the previous draw since the deck is restored to its original state. This is what makes the two events—drawing the first card and drawing the second card—independent.

• If two events are independent, the probability of both occurring is the product of the probabilities of each occurring separately.
• In the context of our card drawing example, because the card is replaced, the probability of drawing an ace both times remains the same.
• Thus, the calculation involves multiplying the probability of drawing an ace once by itself, as both draws do not influence each other.

Understanding independent events is crucial in probability calculations, especially in scenarios involving replacement.

Deck of Cards

A standard deck of cards is commonly used in probability exercises due to its well-defined structure and complexity. Understanding the configuration of a deck is fundamental in solving related probability problems. A typical deck has:

• 52 cards in total.
• 4 suits: Hearts, Diamonds, Clubs, and Spades, each containing 13 cards.
• Among these, there are 4 aces, one of each suit.
• Similarly, there are 13 cards of each suit—a prime consideration when determining probability events such as drawing a spade.

Knowledge of the deck’s structure allows for precise calculations of probabilities, such as determining how likely it is to draw certain combinations of cards.
This forms the basis for solving numerous card-related probability problems, like those involving drawing aces or spades.

Probability Calculation

Calculating probability involves determining how likely an event is to occur out of all possible outcomes. For events involving a standard deck of cards, this process typically involves counting the favorable outcomes over the total possible outcomes. Here's how the calculation plays out:

• For drawing two aces with replacement, calculate the probability of drawing an ace on a single draw, which is \(\frac{4}{52}\), as there are 4 aces in 52 cards. Multiply this by itself for two draws (independent events) to get \(\left(\frac{4}{52}\right) \times \left(\frac{4}{52}\right) = \frac{1}{169}\).
• For drawing an ace and then a spade, determine the probability of the first event (an ace, \(\frac{4}{52}\) ) and the second event (a spade, \(\frac{13}{52}\) ), then multiply them together: \(\frac{4}{52} \times \frac{13}{52} = \frac{1}{52}\).

Always remember that a bigger denominator means more possible outcomes, leading to a smaller probability value, reflecting an event that's less likely to occur.
Through practicing these calculations, students can enhance their capability to handle diverse probability-related problems with confidence.