Problem 15 - Basic Statistics for Business and Economics with Student

Question

Suppose the probability you will get an \(\mathrm{A}\) in this class is .25 and the probability you will get a \(\mathrm{B}\) is \(.50 .\) What is the probability your grade will be above a \(C ?\)

Step-by-Step Solution

Answer

The probability of your grade being above a C is 0.75.

Step 1: Understand the Problem

In this problem, we need to find the probability that the grade obtained is above a C, given the individual probabilities of obtaining an A or a B. We are given: \( P(A) = 0.25 \) and \( P(B) = 0.50 \). Grades above a C include both A and B.

Step 2: Calculate the Total Probability

To find the probability of obtaining either an A or a B, we need to sum their individual probabilities since these two events are mutually exclusive. This can be expressed as: \[ P(A \text{ or } B) = P(A) + P(B) \].

Step 3: Perform the Addition

Substitute the given probabilities into the formula: \[ P(A \text{ or } B) = 0.25 + 0.50 \].

Step 4: Compute the Final Result

Add the probabilities together: \[ P(A \text{ or } B) = 0.75 \]. This is the probability that your grade will be above a C.

Key Concepts

Mutually Exclusive Events

In probability, mutually exclusive events are events that cannot happen at the same time. This means if one event occurs, the other cannot. For instance, in the given exercise, achieving a grade of an A or a B are mutually exclusive events. If you get an A, you cannot simultaneously get a B.
Understanding mutually exclusive events is crucial because it affects how probabilities are calculated. With mutually exclusive events, the probability of either event occurring is simply the sum of their individual probabilities.
For example:

• If the probability of getting an A is 0.25
• And the probability of getting a B is 0.50

The probability of getting either an A or a B is 0.75. This simplifies the calculation and helps us understand the chance of outcomes without overlap.

Addition Rule

The addition rule is a fundamental concept in probability. It allows us to find the probability that any one of several mutually exclusive events occurs. For mutually exclusive events, like getting either an A or a B in your class, the addition rule formula is:
\[ P(A \text{ or } B) = P(A) + P(B) \]
This formula is straightforward when dealing with mutually exclusive events because there's no overlap between them.In our context, we use the addition rule to calculate how likely it is to get an A or a B. Since these grades don't happen simultaneously, adding their probabilities gives the total chance of improving beyond a C.

Grades Analysis

Analyzing grades in terms of probability can provide valuable insights about potential outcomes. In this exercise, we look at how likely you are to score above a C in a class, given probabilities for specific grades: A and B.
The key steps involve:

• Understanding the probability of each grade individually.
• Using these probabilities to analyze overall grade outcomes.

By applying the concept of mutually exclusive events and the addition rule, you can clearly determine that there's a 75% chance of scoring above a C. This analysis can help you focus more effectively on your studies, knowing where potential outcomes lie. Additionally, it offers a strategic approach in academics by estimating future performance scenarios.