Problem 28 - Algebra 2

Question

In a library, the probability that a book is a hardback, given that it is illustrated, is \(0.40 .\) The probability that the book is hardback and illustrated is \(0.20 .\) Find the probability that a book is illustrated.

Step-by-Step Solution

Answer

The probability that a book is illustrated (\(P(Illustrated)\)) is 0.50.

Step 1: Understand Conditional Probability Formula

Understand how the formula for conditional probability works. The formula is \(P(A|B) = \frac{P(A \cap B)}{P(B)}\), where \(P(A|B)\) is the probability of event A occurring given that event B has occurred, \(P(A \cap B)\) is the probability of both A and B occurring, and \(P(B)\) is the probability of event B.

Step 2: Replace The Given in the Formula

Substitute the values given by the problem into the formula. For this problem, the conditional probability \(P(Hardback|Illustrated) = 0.40\), and the joint probability \(P(Hardback \cap Illustrated) = 0.20\). So, the formula becomes \(0.40 = \frac{0.20}{P(Illustrated)}\). The goal is to find \(P(Illustrated)\).

Step 3: Solve for P(Illustrated)

Rearrange the equation to solve for \(P(Illustrated)\). Multiply both sides of the equation by \(P(Illustrated)\) to get \(0.40 * P(Illustrated) = 0.20\). Then, divide both sides by 0.40 to isolate \(P(Illustrated)\). So, \(P(Illustrated) = \frac{0.20}{0.40} = 0.50\).

Key Concepts

Probability of Joint Events

Joint events in probability refer to scenarios where two or more events happen at the same time. These are often denoted using the intersection symbol \( \cap \) in mathematics. When talking about joint events, we might be interested in knowing how often they happen together out of all possible outcomes. To find this, we use the joint probability formula, which is simply the probability of event A and event B occurring at the same time:

• For example, if you're rolling a die, the probability that the roll is both even and a "4" is a joint probability.

In the context of the library problem, when the question talks about the probability of books being both hardback and illustrated, it's referring to a joint event. We are given this probability directly as \( P(Hardback \cap Illustrated) = 0.20 \). This means that 20% of the books are both illustrated and have a hardback cover.
Understanding the probability of joint events is crucial in solving complex probability questions, especially when events are not independent.

Probability of Illustrated Books

The concept of finding the probability of illustrated books in this library problem requires using the information given about other probabilities. Specifically, we use conditional probabilities to determine this.

• Conditional probability refers to the likelihood of event A occurring given that another event B has already occurred.

In our given problem, we have the conditional probability \( P(Hardback|Illustrated) = 0.40 \), meaning that if a book is illustrated, there is a 40% chance it is hardback.
We also know the joint probability \( P(Hardback \cap Illustrated) = 0.20 \), which allows us to use the formula for conditional probability: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
By rearranging the terms, we can solve for \( P(Illustrated) \). As shown in the steps, substituting the known values gives us:
0.40 \( \times \) \( P(Illustrated) = 0.20 \).
Solving this equation yields \( P(Illustrated) = 0.50 \), signifying that half of the books in the library are illustrated.

Probability Theorems

Probability theorems form the backbone of understanding how probabilities are calculated and related to each other. Familiarizing oneself with these theorems helps to solve complex problems more easily.
One of the fundamental theorems is about conditional probability, which tells us how to calculate the likelihood of an event given the occurrence of another event. This is crucial when dealing with dependent events.

• The formula \( P(A|B) = \frac{P(A \cap B)}{P(B)} \) frames the relationship between joint probability and conditional probability.

Another important concept is the multiplication theorem, used when finding the chance of two independent events happening consecutively.
These theorems allow us to connect different pieces of probability information, like how we used the given joint and conditional probabilities to solve for the probability of illustrated books in the original exercise.
Such theorems ensure that calculations are not only correct but also consistent with the mathematical principles of probability.