Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Refer to the following experiment: Two cards are drawn in succession without replacement from a standard deck of 52 cards. What is the probability that the first card is a jack given that the second card is an ace?

Step-by-Step Solution

Answer

The probability that the first card is a Jack given that the second card is an Ace is \(\frac{1}{50}\).

Step 1: Understanding Conditional Probability

Conditional probability is a concept used in probability theory to describe the likelihood of an event occurring, given that another event has occurred. In our case, we want to find the probability that the first card is a jack (event A) given that the second card is an ace (event B). Mathematically, we can write this as: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Step 2: Calculate Individual Probabilities

First, we need to calculate the probabilities of events A and B, and then find the probability of both events happening (A∩B). Event A - Probability that the first card is a Jack: There are 4 Jacks in a deck of 52 cards. \[ P(A) = \frac{4}{52} \] Event B - Probability that the second card is an Ace, given that the first card was a Jack: There will now be 51 cards remaining in the deck and 4 Aces. \[ P(B|A) = \frac{4}{51} \]

Step 3: Calculate Joint Probability (A∩B)

Now we can calculate the probability of both events A and B occurring together (A∩B), which is the probability that the first card is a Jack and the second card is an Ace. We do this by multiplying the probability of A by the probability of B given A: \[ P(A \cap B) = P(A) \times P(B|A) \] \[ P(A \cap B) = \frac{4}{52} \times \frac{4}{51} \]

Step 4: Calculate Probability of B (second card is an Ace)

We finally need to calculate the probability of B, which is the probability that the second card drawn is an Ace. We can do this by splitting the deck into Aces and non-Aces and calculate the individual probabilities. There are two cases for drawing an Ace as a second card: 1. Drawing a non-Ace first and then drawing an Ace: \[P(\text{non-Ace}) \times P(\text{Ace}|\text{non-Ace}) = \frac{48}{52} \times \frac{4}{51}\] 2. Drawing an Ace first and then drawing another Ace: \[P(\text{Ace}) \times P(\text{Ace}|\text{Ace}) = \frac{4}{52} \times \frac{3}{51}\] Now, add both cases probabilities to get the probability of event B: \[P(B) = \frac{48}{52} \times \frac{4}{51} + \frac{4}{52} \times \frac{3}{51} \]

Step 5: Calculate Conditional Probability

Finally, we can find the probability of event A (first card is a Jack) given event B (second card is an Ace) using the conditional probability formula: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] \[ P(A|B) = \frac{\frac{4}{52} \times \frac{4}{51}}{\frac{48}{52} \times \frac{4}{51} + \frac{4}{52} \times \frac{3}{51}} \] Calculate the probabilities and simplify: \[ P(A|B) = \frac{1}{50} \] The probability that the first card is a Jack given that the second card is an Ace is 1/50.

Key Concepts

Probability Theory

Probability theory is a fundamental branch of mathematics that deals with quantifying uncertainty. It allows us to understand and predict outcomes based on known conditions or previous occurrences. In simple terms, probability is a measure of how likely an event is to happen.

In the context of this exercise, probability theory helps us calculate the likelihood of drawing certain cards from a deck. When events are dependent on each other, such as drawing cards without replacement, we use the formula for conditional probability.

The conditional probability formula is given by:

• \( P(A|B) = \frac{P(A \cap B)}{P(B)} \)

This formula finds the probability of event A occurring, given that event B has already occurred.

Deck of Cards

A standard deck of cards is a common tool used to illustrate probability concepts due to its familiarity and well-defined structure.

A standard deck consists of 52 cards divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, ranging from Ace through King, and includes one Jack, one Queen, and one King.

Understanding the composition of a deck is crucial for solving probability problems related to it, like determining the likelihood of certain cards being drawn. In this exercise, properties of the deck are essential for calculating the probability of drawing a Jack first and an Ace second. Without replacement, the number of cards and specific cards left change, influencing subsequent event probabilities.

Joint Probability

Joint probability is a critical aspect of probability theory, referring to the probability of two events happening at the same time. In mathematical terms, it represents the intersection of two events, noted as \( A \cap B \).

For the problem at hand, joint probability helps determine if the first card is a Jack and the second card is an Ace. The formula to calculate this is:

• \( P(A \cap B) = P(A) \times P(B|A) \)

Here, \( P(A) \) is the probability of the first draw being a Jack, and \( P(B|A) \) is the probability of drawing an Ace on the second attempt given that the first card was a Jack.

Understanding joint probability is essential for calculating combined probabilities within dependent events.

Mathematical Notation

Mathematical notation is a system of symbols used to write equations and formulas succinctly. In probability, it aids in expressing complex relationships clearly and concisely.

In the conditional probability formula \( P(A|B) \), the notation indicates the "probability of A given B." Similarly, the equation \( P(A \cap B) \) represents joint probability; it denotes both events A and B occurring together.

Using these symbols, we can solve probability problems more efficiently and clearly, an essential skill in mathematical communication. Understanding notations is crucial to decipher complex problems involving probabilistic events and scenarios, making it an indispensable tool for students working through these concepts.