Problem 44 - College Algebra

Question

The mathematics department of a college has 8 male professor. II female professors, 14 male teaching assistants, and 7 female teaching assistants. If a person is selected at random from the group, find the probability that the selected person is: a professor or a female.

Step-by-Step Solution

Answer

The probability that the selected person is either a professor or a female is \(\frac{13}{20}\).

Step 1: Identify Total Number of People

Sum up the total number of people: 8 male professors, 11 female professors, 14 male teaching assistants, and 7 female teaching assistants. This gives us a total of \(8 + 11 + 14 + 7 = 40\) people.

Step 2: Calculate Probability of Selecting a Professor

Next, sum up the number of professors: \(8 + 11 = 19\) professors in total. The probability of selecting a professor is the ratio of professors to the total number of people, which is \(\frac{19}{40}\).

Step 3: Calculate Probability of Selecting a Female

Next, sum up the number of females: \(11 + 7 = 18\) females in total. The probability of selecting a female is the ratio of females to the total number of people, which is \(\frac{18}{40}\).

Step 4: Adjust for Double Counting

When adding the probabilities from steps 2 and 3, people who are female professors will be counted twice, once in each category. To adjust for this, subtract the probability of selecting a female professor, which is \(\frac{11}{40}\).

Step 5: Calculate total probability

Finally, add adjusted probabilities from step 2, 3 and 4 to get the total probability of selecting a professor or a female. \(\frac{19}{40} + \frac{18}{40} - \frac{11}{40} = \frac{26}{40}\), then simplified to \(\frac{13}{20}\).

Key Concepts

Professors

When considering the probability of selecting a professor at a random from a group, it's crucial to understand how we define this group. Professors, in an educational context, are often considered as educators or instructors with advanced knowledge in their subject area.
In our scenario, we have both male and female professors. The total number of professors is simply the sum of male and female professors in the group.

• For example, we have 8 male professors and 11 female professors.
• Adding these gives us a total of 19 professors.

This total allows us to calculate the probability of selecting someone who is a professor. In probability, you want to find the ratio of the specific group of interest (in this case, professors) compared to the total population. Here, the probability that a person selected is a professor can be calculated by dividing the number of professors by the total number of people, which is \(\frac{19}{40}\).
Understanding the breakdown of the group is essential for such probability calculations.

Female Participants

Identifying the female participants in any scenario helps in understanding gender distribution and related probabilities. Female participants in our scenario include both professors and teaching assistants.

• We have 11 female professors and 7 female teaching assistants.
• This gives us a total of 18 female participants when added together.

The probability of selecting a female participant from the group involves dividing the number of female participants (18) by the total number of people (40). Thus, the probability is \(\frac{18}{40}\).
This simple calculation helps in figuring out how likely it is, in relative terms, for the selected individual to be female. It's essential to count every group member correctly and ensure that no one is double-counted unless specified for adjustments.

Counting Principles

Counting principles are fundamental in probability because they guide us on how to accurately account for the population's subgroups and avoid overlapping counts. When dealing with several categories, as in our problem, it's possible to mistakenly double count members who belong to more than one category.
The principle starts with the summation of all members to find the basic counts for distinct features. For overlapping features, like female professors who fall into both categories, this method requires us to adjust the count to prevent errors.

• First, calculate the total number in each category, such as the total professors or females.
• If a person belongs to both categories, they will initially be counted twice.

Examples from our problem demonstrate this tweak when we are considering professors and female participants. Adjusting for double counts is done by subtracting where these categories overlap - in this case, subtracting the 11 female professors, and then adding the non-overlapping parts.
Such principles ensure we keep our calculations accurate and help us arrive at the correct probabilities.

Teaching Assistants

Teaching assistants play a vital role in educational settings, often providing support to professors and students. In probability assessments, they represent a key subgroup that can affect outcomes if counted accurately.
In our problem, we have both male and female teaching assistants. It is crucial to identify how many there are of each gender and how this subgroup fits into the larger picture of probabilities.

• There are 14 male teaching assistants and 7 female teaching assistants.
• Altogether, they make up a group of 21 teaching assistants.

It helps to see how this subgroup fits in the wider set when determining the likelihood of choosing a particular type of person at random. Although our immediate problem focuses on professors and females, understanding each subgroup's place ensures we count everyone correctly and set the foundation for any expanded probability scenarios involving teaching assistants.