Problem 49 - Precalculus with Limits

Question

A college sends a survey to selected members of the class of 2009. Of the 1254 people who graduated that year, 672 are women, of whom 124 went on to graduate school. Of the 582 male graduates, 198 went on to graduate school. An alumni member is selected at random. What are the probabilities that the person is (a) female, (b) male, and (c) female and did not attend graduate school?

Step-by-Step Solution

Answer

The probability that the selected person is a female is \( \frac{672}{1254} \), the probability that the selected person is a male is \( \frac{582}{1254} \), and the probability that the selected woman did not attend graduate school is \( \frac{548}{1254} \).

Step 1: Calculate the Probability of a Person Being Female

To calculate this probability, we use the formula \[ P(A) = \frac{Number \ of \ favourable \ outcomes}{Total \ number \ of \ outcomes} \] Here, Number of favourable outcomes = Number of females = 672 and Total number of outcomes = Total number of alumni = 1254 Substituting these values in the formula \[ P(A) = \frac{672}{1254} \]

Step 2: Calculate the Probability of a Person Being Male

To calculate this probability, we use the same formula \[ P(B) = \frac{Number \ of \ favourable \ outcomes}{Total \ number \ of \ outcomes} \]Number of favourable outcomes = Number of males = 582 Total number of outcomes = Total number of alumni = 1254 Substituting these values in the formula \[ P(B) = \frac{582}{1254} \]

Step 3: Calculate the Probability of a Woman Not Going to Graduate School

To calculate this probability, we use the same probability formula. Number of favourable outcomes = Number of females who didn't go to grad school = Total females - Females who went to grad school = 672 - 124 = 548 Total number of outcomes = Total number of alumni = 1254 Substituting these values in the formula \[ P(C) = \frac{548}{1254} \]

Key Concepts

Probability Formula

Understanding the probability formula is crucial when beginning to study probability, which is a measure of the likelihood that an event will occur. The formula can be expressed simply as:
\[\begin{equation} P(A) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}} \end{equation}\]
With this, probability values range from 0 to 1, where 0 indicates impossibility and 1 signifies certainty. In the context of the original exercise, where you're asked about the likelihood of selecting a female alumni member, it becomes a straightforward calculation. Here, the number of favorable outcomes is the number of female graduates, and the total number of outcomes is the total number of alumni. Similarly, this formula also applies when determining the probability of selecting a male graduate or a female who didn't attend graduate school. One common aspect students often overlook is the necessity to clearly define favorable and total outcomes to avoid confusion and ensure an accurate calculation.

Probability Calculation

Breaking down a probability calculation can make it more digestible for students, especially when dealing with compound events, which involve the combination of two or more events. In our example, to calculate the probability of a female graduate who did not attend graduate school (\[\begin{equation} P(C) = \frac{\text{Number of females} - \text{Females who went to grad school}}{\text{Total number of alumni}} \end{equation}\]), you subtract the number of females who went to graduate school from the total number of female graduates. The result becomes the numerator in the probability formula, with the denominator remaining the total alumni count. This illustrates an important principle: for accurate probability calculation, always update your favorable outcomes based on the specific criteria of the event in question.

Clarity in Event Definition

It's important to be precise in defining the events for which you're calculating the probability. Misunderstanding the criteria for a favorable outcome can lead to incorrect probabilities. The clearer the event definition, the more accurate the probability calculation.

Graduate School Statistics

When stepping up to graduate-level studies, students encounter statistics as both a tool and a subject of study. Graduate school statistics often require a deeper understanding of probability and its applications. In our alumni example, graduate school likelihood for different genders might be a subject of statistical analysis to understand trends in higher education.

Real-World Implications

Studying graduate school statistics sheds light on patterns and decision-making processes within education and professional spheres. By examining actual data, like the number of male and female graduates pursuing further education, students can learn to produce and interpret statistical evidence. This critical analysis is pivotal in various fields, including government policy, academic research, and market research, where probabilities and statistics inform significant decisions. Consequently, real-world exercises such as the alumni survey provide practical experience in the kind of statistical reasoning that graduate programs and employers value.