Problem 20 - Algebra 2

Question

For Exercises \(20-23,\) determine whether the events are mutually exclusive or inclusive. Then find the probability. There are 4 algebra books, 3 literature books, and 2 biology books on a shelf. If a book is randomly selected, what is the probability of selecting a literature book or an algebra book?

Step-by-Step Solution

Answer

The probability of selecting a literature or algebra book is \( \frac{7}{9} \).

Step 1: Identify Event Types

We have two events to consider: selecting a literature book and selecting an algebra book. We need to determine whether these events are mutually exclusive or not (i.e., do they have any common outcomes). Since a book cannot be both a literature and algebra book simultaneously, the events are mutually exclusive.

Step 2: Determine Total Number of Books

Add up the total number of books on the shelf. There are 4 algebra books, 3 literature books, and 2 biology books, totaling: 4 + 3 + 2 = 9 books.

Step 3: Calculate Probability of Each Event

Calculate the probability of selecting a literature book: Probability of literature book = \( \frac{3}{9} = \frac{1}{3} \).Calculate the probability of selecting an algebra book: Probability of algebra book = \( \frac{4}{9} \).

Step 4: Use Addition Rule for Mutually Exclusive Events

Since the events are mutually exclusive, use the addition rule to find the probability of either event occurring. Sum the individual probabilities: Probability(literature or algebra) = \( \frac{1}{3} + \frac{4}{9} = \frac{3}{9} + \frac{4}{9} = \frac{7}{9} \).

Key Concepts

Mutually Exclusive Events

In probability theory, the term 'mutually exclusive events' refers to scenarios where two events cannot occur simultaneously. Imagine having a single book from a shelf that needs to be picked, and you are interested in choosing either an algebra book or a literature book. The nature of the books dictates that it can either be an algebra book or a literature book, but not both at the same time. Here's why this concept is important:

• It helps determine how we calculate probabilities. If events are mutually exclusive, it simplifies things because the occurrence of one event means the non-occurrence of the other.
• Understanding mutually exclusive events helps in applying the correct probability rules, especially the addition rule for probabilities.

Using this concept ensures clarity when assessing the probability of combined outcomes, such as in our exercise where picking an algebra book excludes the possibility of getting a literature book.

Addition Rule for Probabilities

The addition rule for probabilities is incredibly useful, especially when dealing with mutually exclusive events. It helps us find the probability of one event or another occurring. When events are mutually exclusive, like in our exercise, the addition rule becomes straightforward and highly applicable.Consider two events, A and B, that are mutually exclusive:- Event A: Choosing a literature book- Event B: Choosing an algebra bookThe probability of A or B happening can be calculated using the formula:\[P(A \text{ or } B) = P(A) + P(B)\]In simpler terms:

• Add the individual probabilities of each event.
• No need to worry about subtracting any overlap since mutually exclusive events have none.

By applying the addition rule from our example, we sum the probabilities of selecting either a literature book or an algebra book to yield a clear result.

Probability Calculation

Calculating probabilities involves determining how likely an event is to occur out of the total possible outcomes. For our bookshelf example, we needed to calculate the probabilities of selecting specific types of books.Here's how we calculated them:

• First, determine the total number of outcomes. In our case, there are 9 total books.
• Next, find the favorable outcomes for each event, such as 3 literature books and 4 algebra books.
• Calculate each probability separately:
  • Probability of a literature book is \( \frac{3}{9} = \frac{1}{3} \)
  • Probability of an algebra book is \( \frac{4}{9} \)

Once individual probabilities are found, use the addition rule to determine the overall probability if events are mutually exclusive, which in this case, was found to be \( \frac{7}{9} \). This calculation tells us the likelihood of one of these two events happening when pulling a book from the shelf at random.