Problem 20 - Basic Statistics for Business and Economics with Student

Question

A student is taking two courses, history and math. The probability the student will pass the history course is \(.60,\) and the probability of passing the math course is \(.70 .\) The probability of passing both is .50. What is the probability of passing at least one?

Step-by-Step Solution

Answer

The probability of passing at least one course is 0.80.

Step 1: Understanding the Problem

The problem asks us to find the probability that the student passes at least one of the two courses, history or math. We are given the probabilities of passing each course separately and the probability of passing both courses.

Step 2: Using the Formula for Probability of Union

To find the probability of passing at least one course, we can use the formula for the probability of the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] where \(P(A)\) is the probability of passing history, \(P(B)\) is the probability of passing math, and \(P(A \cap B)\) is the probability of passing both courses.

Step 3: Substituting Given Values

Substitute the given probabilities into the formula: \[ P(A \cup B) = 0.60 + 0.70 - 0.50 \]

Step 4: Calculating the Result

Calculate the expression: \[ 0.60 + 0.70 - 0.50 = 0.80 \] Therefore, the probability of passing at least one course is 0.80.

Key Concepts

Probability Calculations

Probability calculations are a fundamental aspect of probability theory. They help us quantify the likelihood of different outcomes occurring. In our exercise, the task is to find the probability of at least one event happening, meaning the student passes either the math or the history course. Here, we use known probabilities to calculate an unknown probability.

There are some basic components you need to understand when working with probability calculations:

• Probability of a single event (\( P \)): This measures the likelihood of a particular result occurring. In our exercise, \( P(A) = 0.60 \) for passing history, and \( P(B) = 0.70 \) for passing math.
• Joint probability (\( P(A \cap B) \)): This represents the probability of two events happening simultaneously, in our case, passing both courses at \( P(A \cap B) = 0.50 \).
• Union probability (\( P(A \cup B) \)): This reflects the probability of at least one of the events occurring, calculated using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \].

Being precise in these calculations is crucial to getting accurate results. Understanding these concepts is key when calculating occurrences in probability theory.

Union of Events

The union of events is a core idea in probability that allows us to calculate the probability of multiple events happening independently or together. In simpler terms, it's the likelihood that at least one of the events occurs.

In our exercise, when asked for the probability that a student passes at least one course, we apply the notion of union to define events "pass history" and "pass math" as potential outcomes. This can be represented as:

• Union Formula: As demonstrated, \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] helps compute the union of both events. By summing the individual probabilities and subtracting the overlap, we ensure we don't count students passing both courses twice.

This concept is not only important for this specific exercise but also widely applicable to various scenarios in data analysis, risk assessment, and decision-making processes involving probabilities.

Basic Statistics

Basic statistics is the foundational toolset for analyzing data, understanding patterns, and predicting outcomes. It includes concepts like probability, which quantifies uncertainty and helps in making informed guesses or predictions based on data.

In probability theory, basic statistics involve:

• Events: Any situation or outcome we can measure as happening or not, like passing a course.
• Sample space: The entire range of possible outcomes. For example, passing or failing each course would be part of this.
• Probability distributions: These show how likely different outcomes are, crucial for understanding what might happen under different conditions.

By learning the basics of statistics, you can better comprehend details of probability calculations and efficiently solve problems like the one stated in our exercise. Doing so not only boosts your math skills but also enhances your ability to apply logic to real-world situations.