Problem 27 - Calculus for Biology and Medicine

Question

You are dealt 1 card from a standard deck of 52 cards. If \(A\) denotes the event that the card is a spade and if \(B\) denotes the event that the card is an ace, determine whether \(A\) and \(B\) are independent.

Step-by-Step Solution

Answer

Events A and B are independent because \( P(A \cap B) \) equals \( P(A) \cdot P(B) \).

Step 1: Understand the Definitions

Two events, A and B, are independent if the probability of both events occurring is the product of their individual probabilities. Mathematically, this is expressed as \( P(A \cap B) = P(A) \cdot P(B) \). We will use this definition to determine if events A and B are independent.

Step 2: Find the Probability of A

Event A is the probability that the card is a spade. In a standard deck, there are 13 spades. Therefore, \( P(A) = \frac{13}{52} = \frac{1}{4} \).

Step 3: Find the Probability of B

Event B is the probability that the card is an ace. In a standard deck, there are 4 aces. Therefore, \( P(B) = \frac{4}{52} = \frac{1}{13} \).

Step 4: Find the Probability of A Intersection B

Event \( A \cap B \) is the probability that the card is both a spade and an ace, which is the ace of spades. Since there is only one such card, \( P(A \cap B) = \frac{1}{52} \).

Step 5: Check for Independence

For A and B to be independent, \( P(A \cap B) \) should equal \( P(A) \cdot P(B) \). Calculate \( P(A) \cdot P(B) = \frac{1}{4} \times \frac{1}{13} = \frac{1}{52} \). Compare \( P(A \cap B) = \frac{1}{52} \) with this result.

Step 6: Conclude Independence

Since \( P(A \cap B) = \frac{1}{52} \) and \( P(A) \cdot P(B) = \frac{1}{52} \), A and B are indeed independent events.

Key Concepts

Independent Events

In the world of probability, understanding independent events is crucial. Two events, say, Event A and Event B, are known as independent if the occurrence of one does not affect the occurrence of the other. Mathematically, this relationship is expressed through the formula:

• If and only if Event A and Event B are independent, then the probability of both events happening simultaneously (denoted as \( P(A \cap B) \)) is equal to the product of their individual probabilities: \( P(A) \cdot P(B) \).
• This means that the joint probability of both events occurring should exactly match the expected result from multiplying their separate probabilities.

To visualize, imagine drawing a card from a deck: the chance of drawing a spade should not influence the chance of drawing an ace. If multiplying these probabilities matches the joint probability, the events are independent. This concept helps in simplifying and understanding complex probability scenarios.

Deck of Cards

A standard deck of cards is a wonderful tool for learning about probability. It contains 52 cards, with each card showing one of four suits: spades, hearts, diamonds, and clubs. Here’s a breakdown:

• Each suit has 13 cards, numbered from 2 to 10, and including a jack, queen, king, and an ace.
• Thus, there are 13 spades, 13 hearts, 13 diamonds, and 13 clubs in total.
• The deck also includes 4 aces, one from each suit.

Working with a deck of cards is not just fun, but also helps conceptualize probability. It provides clear and tangible examples to calculate different probabilities, such as the chance of drawing a particular suit or face card. For instance, in the exercise at hand, we examine the events "drawing a spade" and "drawing an ace," directly from this deck.

Probability Calculation

Calculating probability involves analyzing the likelihood of an event occurring. In probability theory, we often calculate simple probabilities or assess if events are independent. Let's work through a simple calculation step.

• Probability of an event is defined as the number of favorable outcomes divided by the total number of possible outcomes (\( P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}} \)).
• Using our deck of cards, to find the probability of drawing a spade, we count 13 spades out of 52 cards, giving us \( P(A) = \frac{13}{52} = \frac{1}{4} \).
• Similarly, for drawing an ace, there are 4 aces. Hence, \( P(B) = \frac{4}{52} = \frac{1}{13} \).

For assessing independence, we check if the probability of both events occurring (A spade that is also an ace) equals the product of their separate probabilities. If true, it confirms their independence.