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

Yes, events A and B are independent.

Step 1: Understand the Definitions

Two events, A and B, are independent if the occurrence of one does not affect the probability of the other. Mathematically, two events are independent if \( P(A \cap B) = P(A) \cdot P(B) \).

Step 2: Calculate Probability of A

Event A is the event that you draw a spade. There are 13 spades in a deck of 52 cards. Thus, the probability of drawing a spade is \( P(A) = \frac{13}{52} = \frac{1}{4} \).

Step 3: Calculate Probability of B

Event B is the event that you draw an ace. There are 4 aces in a deck of 52 cards. Thus, the probability of drawing an ace is \( P(B) = \frac{4}{52} = \frac{1}{13} \).

Step 4: Calculate Probability of A and B

The event \( A \cap B \) is the scenario where the card is both a spade and an ace. There is only 1 card, the Ace of Spades, that satisfies this condition. Thus, \( P(A \cap B) = \frac{1}{52} \).

Step 5: Check Independence

To determine if events A and B are independent, check whether \( P(A \cap B) = P(A) \cdot P(B) \). We know \( P(A \cap B) = \frac{1}{52} \), and \( P(A) \cdot P(B) = \frac{1}{4} \cdot \frac{1}{13} = \frac{1}{52} \). Since these probabilities are equal, A and B are independent.

Key Concepts

Independent Events

Independent events are fundamental in probability theory. Imagine you're flipping a coin and rolling a die. The outcome of the coin doesn't affect the die's outcome and vice versa. That's the gist of independent events. Two events, say event A and event B, are independent if the outcome of one event does not influence the occurrence of the other.
For example, in a deck of playing cards, imagine you pick a card. Let's say event A is drawing a spade and event B is drawing an ace. They are considered independent if knowing the outcome of one doesn't change the probability of the other occurring. Mathematically, this is expressed as: \( P(A \cap B) = P(A) \cdot P(B) \). In simple terms:

• The probability of drawing a spade from a deck (Event A) = \( \frac{1}{4} \) as there are 13 spades.
• The probability of drawing an ace (Event B) = \( \frac{1}{13} \) since there are 4 aces in a deck.

Independent events will show that the combined probability \( P(A \cap B) = \frac{1}{52} \) equals \( P(A) \cdot P(B) \). Here, both are indeed equal, confirming independence.

Conditional Probability

Conditional probability deals with finding the probability of an event occurring given that another event has already happened. It shifts our focus slightly because it makes us consider scenarios where some information is already known. For instance:

• Let's say you're trying to find the probability of drawing a spade given that you know the card is an ace.
• It changes our "universe" of options from 52 cards to just 4 aces.
• The probability then becomes finding how many of those aces are spades — and since there's only one ace of spades, the conditional probability becomes 25% or \( \frac{1}{4} \).

Conditional probability is crucial to determine dependencies between events, which then informs decisions and assessments in real-world scenarios. Offering a more calculated perspective, it informs us if assumptions about independence were incorrect.

Probability Calculations

Probability calculations form the bedrock of probability theory. On the surface, probability tells us how likely an event is to happen. The core principles involve:

• Understanding "possible outcomes" versus "favorable outcomes".
• Overall probability is calculated as the number of favorable outcomes divided by the number of possible outcomes.

Taking the previous example:

• To find the probability of drawing a spade (Event A), you take the 13 spades over the 52 cards, resulting in \( \frac{1}{4} \).
• For finding the probability of drawing an ace (Event B), it's 4 aces over 52 cards or \( \frac{1}{13} \).
• If these events overlap, such as drawing an ace of spades (both A and B), it would be \( \frac{1}{52} \).

Probability calculations allow us not only to deduce basic scenarios but also to tackle more complex real-world situations with clarity and accuracy.