Question

If you draw a card at random from a wellshuffled deck, is getting an ace independent of the suit? Explain.

Short answer

Yes, drawing an ace is independent of the suit, as the probabilities are equal.

Step-by-step solution

Understanding Independence of Events

Two events are independent if the occurrence of one event does not affect the occurrence of the other. In this case, we need to determine if drawing an ace from the deck is independent of drawing any particular suit.

Probability of Drawing an Ace

Calculate the probability of drawing an ace from a deck. A deck has 52 cards, and there are 4 aces. Thus, the probability of drawing an ace is \( P(A) = \frac{4}{52} = \frac{1}{13} \).

Probability of Drawing a Suit

Calculate the probability of drawing a particular suit (e.g., hearts) from the deck. There are 13 cards per suit in a deck of 52 cards. Thus, the probability of drawing a particular suit is \( P(S) = \frac{13}{52} = \frac{1}{4} \).

Joint Probability of Drawing an Ace of a Particular Suit

Calculate the probability of drawing both an ace and a card from a specific suit. There is 1 ace per suit. So, the probability is \( P(A \cap S) = \frac{1}{52} \).

Test for Independence

To test for independence, check if the probability of drawing an ace given a particular suit is equal to the probability of drawing an ace independently. Calculate \( P(A|S) \), which is the probability of drawing an ace when we know the suit is fixed. Using the definition of conditional probability: \[ P(A|S) = \frac{P(A \cap S)}{P(S)} = \frac{\frac{1}{52}}{\frac{1}{4}} = \frac{1}{13} \].Since \( P(A) = \frac{1}{13} \) and \( P(A|S) = \frac{1}{13} \), drawing an ace is independent of the suit.

Key concepts

Independence of Events

In probability, the concept of independence helps us understand how the occurrence of one event relates to another. Two events are considered independent when the outcome of one does not affect the outcome of the other.
This means that knowing the result of one event provides no information about the other.

For example, when determining if drawing an ace from a deck is independent of getting a particular suit, independence implies that the likelihood of drawing an ace remains unchanged regardless of the suit.
This can be confirmed by comparing the probabilities involved;

• If the probability of drawing an ace (\(P(A)\)) equals the probability of drawing an ace given a specific suit (\(P(A|S)\)), then these events are independent.
• If these probabilities differ, there is dependence between the events.

Understanding independence is crucial in probability as it ensures our calculations and predictions about combined events are accurate.

Conditional probability

Conditional probability allows us to calculate the probability of an event occurring given that another event has already occurred.
This approach is particularly useful when we want to explore relationships between events or refine our understanding based on existing information.

For instance, consider the probability of drawing an ace if we know the card comes from a specific suit such as hearts. This is expressed as \(P(A|S)\), the probability of an ace occurring given a predefined suit.
The formula for conditional probability is:

• \[P(A|S) = \frac{P(A \cap S)}{P(S)}\]

This equation tells us how to adjust the basic probability of an event by incorporating new information about another event.
In essence, conditional probability narrows the sample space by focusing on the subset relevant to both events, providing a more precise probability.

Deck of Cards Analysis

A deck of cards offers a standardized set of probabilities, making it an excellent tool for understanding basic probability concepts.
Each standard deck contains 52 cards divided evenly across four suits: hearts, diamonds, clubs, and spades.
Each suit holds 13 cards ranging from Ace to King.

Analyzing a deck of cards can illustrate numerous probability concepts. For our exercise:

• The probability of drawing any ace (without regard to suit) is calculated by dividing the number of aces, 4, by the total number of cards, 52. Hence, \(P(A) = \frac{4}{52} = \frac{1}{13}\).
• The probability of drawing a card from a specific suit is the number of cards per suit, 13, divided by the total number of cards. Thus, \(P(S) = \frac{13}{52} = \frac{1}{4}\).
• For joint probability such as drawing an ace of a particular suit like the Ace of Hearts, the probability is \(P(A \cap S) = \frac{1}{52}\).

Deck of cards analysis incorporates these calculations to help explain how probability functions in a controlled, known setting, facilitating a deeper understanding of event probability and interdependencies.