Question

In a large Introductory Statistics lecture hall, the professor reports that \(55 \%\) of the students enrolled have never taken a Calculus course, \(32 \%\) have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. What is the probability that the first groupmate you meet has studied a) two or more semesters of Calculus? b) some Calculus? c) no more than one semester of Calculus?

Short answer

a) 13% b) 45% c) 87%

Step-by-step solution

Understand the Information

From the problem, we know that the students are divided into three groups based on their Calculus experience. \(55\%\) of students have never taken any Calculus course, \(32\%\) have taken only one semester, and the rest have taken two or more semesters.

Calculate the Probability for Two or More Semesters

The total percentage must add up to \(100\%\). We already know \(55\%\) have never taken Calculus and \(32\%\) have taken only one semester. Therefore, the probability that a student has taken two or more semesters is \(100\% - 55\% - 32\% = 13\%\).

Calculate the Probability for Some Calculus

Some Calculus means the student has either taken one semester or two or more semesters. Combine these percentages: \(32\%\) have taken one semester and \(13\%\) have taken two or more semesters. The probability for some Calculus is \(32\% + 13\% = 45\%\).

Calculate the Probability for No More Than One Semester

No more than one semester means the student has taken either zero semesters or one semester. This combines \(55\%\) who have never taken Calculus and \(32\%\) who have taken one semester. The probability for no more than one semester is \(55\% + 32\% = 87\%\).

Key concepts

Probability

Probability is a branch of mathematics that deals with calculating the likelihood of a given event's occurrence, expressed as a number between 0 and 1. In our exercise, we are asked to find the probability of different scenarios related to students’ Calculus experience.

When interpreting probabilities, it's essential to understand that lower values (close to 0) imply an event is less likely to occur, while higher values (close to 1) suggest it's more likely. For instance, in scenario (a) from the exercise, we calculate the probability that a student has taken two or more semesters of Calculus. We achieve this by combining given probabilities and subtracting from the total possible outcome, which is always 100%.

• Probability of taking two or more semesters = 13%
• Probability of taking some Calculus = 45%
• Probability of taking no more than one semester = 87%

These probabilities help in understanding the distribution of Calculus experience among the students in the lecture hall.

Calculus Experience

Calculus Experience in this context refers to how much Calculus coursework a student has completed. Understanding the different levels of expertise is crucial in statistical exercises like this one. Students are categorized based on whether they have taken no Calculus, one semester, or two or more semesters.

This classification helps determine the probability distributions utilized for understanding group dynamics in assignments. In our classroom scenario, understanding the percentage split between these categories allows the professor to predict the skill levels available in each group efficiently.

• 55% have no Calculus experience, indicating a majority likely lack advanced mathematics skills.
• 32% have taken one semester, suggesting moderate familiarity with Calculus concepts.
• 13% have taken two or more semesters, denoting a smaller group with more comprehensive Calculus knowledge.

This distribution is key in predictable group performance, important for seamlessly assigning roles within group assignments.

Group Assignments

Group assignments are integral to collaborative learning, particularly in large classroom settings. They allow students to share knowledge and skills, contributing to each other's learning experiences.

For successful group assignments, understanding the composition of skills within groups is vital. Our exercise demonstrates this by asking how Calculus experience might be distributed in groups of three, randomly selected from a diverse classroom.

When forming groups, the professor might hope to balance these skills to prevent over-burdening or underutilizing certain students' abilities. The probabilities calculated before can inform this process, ensuring a mix of experienced and inexperienced members, promoting stronger outcomes for projects. Balancing groups can:

• Enhance learning through peer instruction.
• Encourage collaborative problem-solving.
• Distribute workload equitably among group members.

This structured approach to group work, supported by calculated probabilities, fosters an environment where students can benefit academically and develop team-working skills.