Problem 60 - 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 is a spade?

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem Context

We have a standard deck of 52 cards. Each card drawn is replaced, meaning the total number of cards remains constant after each draw. We need to calculate probabilities for drawing certain types of cards in sequence.

Step 2: Calculating Probability of Drawing an Ace

There are 4 aces in a deck of 52 cards. The probability of drawing an ace on a single draw is given by dividing the number of aces by the total number of cards: \( P(\text{Ace}) = \frac{4}{52} = \frac{1}{13} \).

Step 3: Probability that Both Cards are Aces

Since the card is replaced, drawing each card is an independent event. Therefore, the probability of drawing an ace twice is the product of the probabilities of drawing an ace each time: \( P(\text{Two Aces}) = \left(\frac{1}{13}\right) \times \left(\frac{1}{13}\right) = \frac{1}{169} \).

Step 4: Calculating Probability that First is an Ace, Second is a Spade

First, we calculate the probability for the sequence: *first an ace*, then *a spade*. There are 13 spades in a deck, so \( P(\text{Spade after Ace}) = \frac{13}{52} = \frac{1}{4} \). The overall probability for this sequence is \( P(\text{Ace then Spade}) = \frac{1}{13} \times \frac{1}{4} = \frac{1}{52} \).

Key Concepts

Understanding Independent Events in Probability

In probability, independent events are those whose outcomes do not affect each other. When you replace a card in a deck after drawing it, each draw is independent. This means that the result of the first draw has no impact on the next. Independent events are critical in probability calculations because they allow us to multiply individual probabilities to get the probability of combined events. For instance, considering the problem of drawing two aces from a deck, replacing the first card ensures that both draws are independent activities. Thus, you multiply the probability of drawing an ace on the first draw by the probability of drawing an ace on the second draw. When events are independent, it simplifies the process of calculating probabilities for sequences of events.

Exploring a Standard Deck of Cards

A standard deck of cards consists of 52 cards. These cards are divided into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, including numbered cards from 2 to 10, and the face cards: jack, queen, king, and ace. Each suit is equally represented, which means you have four aces and four of any other value across different suits. Understanding the composition of a deck is essential for probability calculations, as it dictates the likelihood of drawing any particular card or type of card. For example, since there are four aces in a deck, the initial probability of drawing an ace is calculated as 4 out of 52. This foundational knowledge helps you to tackle more complex probability problems by knowing your 'sample space' or the range of possible outcomes.

The Basics of Probability Calculation

Probability is a way to quantify the likelihood that a certain event will occur. It is calculated as the ratio of favorable outcomes to the total number of possible outcomes. For example, the probability of drawing an ace from a standard deck is calculated as the number of aces divided by the total number of cards: \( P(\text{Ace}) = \frac{4}{52} = \frac{1}{13} \). In the case of multiple independent events, like drawing two cards with replacement, you multiply the probabilities of each individual event. For instance, drawing two aces in a row with replacement has a combined probability of \( \frac{1}{13} \times \frac{1}{13} = \frac{1}{169} \). By multiplying these independent probabilities, you get the overall likelihood of both events happening consecutively.

The Impact of Replacement in Probability

In probability calculations, replacement significantly affects the outcome because it maintains the original conditions of the sample space. When drawing cards from a deck and replacing each one before drawing the next, you keep the total number of cards the same. This means the probability of drawing a specific card type remains unchanged for subsequent draws. Without replacement, the number of each type of card changes with each draw, altering probabilities and making calculations more complex. In the exercise provided, replacement ensures that drawing an ace on the first try doesn't change the number of aces available for the second draw, which keeps the probabilities consistent. This method of drawing helps to simplify problems by ensuring each selection is an independent event, unaffected by previous draws.