Problem 28 - Calculus for Biology and Medicine

Question

You are dealt 2 cards from a standard deck of 52 cards. If \(A\) denotes the event that the first card is an ace and \(B\) denotes the event that the second card is an ace, determine whether \(A\) and \(B\) are independent.

Step-by-Step Solution

Answer

Events \(A\) and \(B\) are not independent because \(P(B|A) \neq P(B)\).

Step 1: Understanding the Problem

We need to determine whether the two events, drawing an ace first and drawing an ace second, are independent. Two events are independent if the occurrence of one does not affect the probability of the other.

Step 2: Calculate Probability of Event A

The event \(A\) is drawing an ace first. There are 4 aces in a deck of 52 cards. Thus, the probability \(P(A)\) is calculated as follows: \[ P(A) = \frac{4}{52} = \frac{1}{13}. \]

Step 3: Calculate Probability of Event B Given A

The event \(B\) is drawing an ace second given that the first card is an ace. After drawing the first ace, there are 3 aces left and only 51 cards in total. Thus, the probability \(P(B|A)\) is: \[ P(B|A) = \frac{3}{51} = \frac{1}{17}. \]

Step 4: Calculate Probability of Event B

To find if \(A\) and \(B\) are independent, we should also consider \(P(B)\), the probability that the second card is an ace without any condition. \[ P(B) = \frac{4}{52} = \frac{1}{13}. \]

Step 5: Determine Independence

Events \(A\) and \(B\) are independent if \(P(B|A) = P(B)\). We found \(P(B|A) = \frac{1}{17}\) and \(P(B) = \frac{1}{13}\). These probabilities are not equal, so events \(A\) and \(B\) are not independent.

Key Concepts

Probability Calculation

Probability is the measure of the likelihood that an event will occur. When it comes to card games, calculating probabilities can make you a more strategic player. The basic formula for calculating the probability of a single event is: \[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}. \] For example, if you want to find the probability of drawing an ace from a standard deck of 52 cards, you calculate it based on the four aces available:\[ P(\text{Ace}) = \frac{4}{52} = \frac{1}{13}. \] Using this same formula, any event can have its probability calculated, as long as you know the number of favorable outcomes and the total possible outcomes.

Conditional Probability

Conditional probability is a vital concept in probability theory that deals with finding the probability of an event given that another event has already occurred. This concept modifies the basic probability by including prior condition information. The formula for conditional probability can be expressed as:\[ P(B|A) = \frac{P(A \cap B)}{P(A)}. \] This formula can be simplified for specific cases. In our exercise, we considered if the event of drawing a second ace (Event B) was still likely once the first ace had been drawn (Event A). The remaining number of cards and aces affects this calculation, thus:\[ P(B|A) = \frac{3}{51} = \frac{1}{17}. \] Conditional probability is crucial for understanding how related events impact one another.

Probability Theory

Probability theory forms the backbone of understanding uncertainty in various fields such as mathematics, engineering, and science. It provides a framework for analyzing random phenomena and predicting the likelihood of events.One fundamental part of probability theory is understanding the independence of events. In probability, two events are independent if the occurrence of one does not affect the probability of the other. This can be checked by ensuring:\[ P(B|A) = P(B). \] In our scenario with drawing cards, after deriving the individual probabilities and the conditional probability, we proved that drawing the second ace (Event B) isn't independent of the first ace (Event A) since:\[ P(B|A) = \frac{1}{17} \quad \text{and} \quad P(B) = \frac{1}{13}. \] As these probabilities are not equal, it implies a dependency, showcasing an important aspect of how probability theory aids in analyzing real-world scenarios.