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The American Council of Education reported that \(47 \%\) of college freshmen earn a degree and graduate within five years (Associated Press, May 6,2002 ). Assume that graduation records show women make up \(50 \%\) of the students who graduated within five years, but only \(45 \%\) of the students who did not graduate within five years. The students who had not graduated within five years either dropped out or were still working on their degrees. a. \(\quad\) Let \(A_{1}=\) the student graduated within five years \(A_{2}=\) the student did not graduate within five years \(W=\) the student is a female student Using the given information, what are the values for \(P\left(A_{1}\right), P\left(A_{2}\right), P\left(W | A_{1}\right),\) and \\[ P\left(W | A_{2}\right) ? \\] b. What is the probability that a female student will graduate within five years? c. What is the probability that a male student will graduate within five years? d. Given the preceding results, what are the percentage of women and the percentage of men in the entering freshman class?

Step by step solution

01

Understand the Problem

We have to find various conditional and unconditional probabilities related to college graduation and gender. We will use probability laws and the given percentages to determine these values.

02

Identify Given Probabilities

The problem states: - \(P(A_1) = 0.47\): the probability a student graduates within five years.- \(P(A_2) = 1 - P(A_1) = 0.53\): the probability a student does not graduate within five years.- \(P(W | A_1) = 0.50\): the probability a student is female given they graduated.- \(P(W | A_2) = 0.45\): the probability a student is female given they did not graduate.

03

Calculate Probability Female Graduates

Using conditional probability, we want \(P(A_1|W)\). By Bayes' theorem, we have:\[P(A_1 | W) = \frac{P(W | A_1) \cdot P(A_1)}{P(W)}\]We must first calculate \(P(W)\) using the law of total probability:\[P(W) = P(W | A_1) \cdot P(A_1) + P(W | A_2) \cdot P(A_2)\]\(P(W) = 0.50 \times 0.47 + 0.45 \times 0.53 = 0.235 + 0.2385 = 0.4735\).Now substitute back:\[P(A_1 | W) = \frac{0.50 \times 0.47}{0.4735} = \frac{0.235}{0.4735} \approx 0.496\]Thus, the probability a female graduates within five years is approximately 0.496 or 49.6%.

04

Calculate Probability Male Graduates

To find \(P(A_1 | M)\) where \(M\) is male, note that \(P(M) = 1 - P(W) = 0.5265\). Using logic similar to Step 3, we apply the complement:\[P(A_1 | M) = \frac{P(M | A_1) \cdot P(A_1)}{P(M)}\]Since \(P(M | A_1) = 1 - P(W | A_1) = 0.50\):\[P(A_1 | M) = \frac{0.50 \times 0.47}{0.5265} = \frac{0.235}{0.5265} \approx 0.446\]Thus, the probability a male graduates within five years is approximately 0.446 or 44.6%.

05

Calculate Percentage of Women and Men in Freshman Class

Using the previous results and initial data, consider the total probability and ratios:- The percentage of women in the class is the same as the calculated \(P(W)\) from Step 3, which is 47.35%.- The percentage of men is the remaining portion, which is 52.65%, calculated as \(1 - P(W)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bayes' theorem

Here we'll explore how Bayes' theorem helps solve conditional probability problems. Bayes’ theorem is a mathematical formula that allows us to update the probability of a hypothesis based on new evidence. It's incredibly useful for calculating probabilities with conditional data. For instance, in the context of gender statistics in graduation rates, Bayes' theorem computes the probability a female student will graduate when we know certain gender statistics.
The formula for Bayes' theorem is:

• \(P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)}\)

In this equation:

• \(P(A | B)\) is the probability of event A occurring given event B is true.
• \(P(B | A)\) is the probability of event B given event A.
• \(P(A)\) and \(P(B)\) are the probabilities of A and B independently occurring.

By applying Bayes' theorem, we can determine the conditional probability that a female student graduates within five years from the given statistics.

probability laws

Probability laws are foundational rules that apply to probability calculations. The most important ones from the exercise are the complement rule and the law of total probability. These rules help break down complex probability questions into simpler components:

• The **Complement Rule** states that the probability of an event not occurring is 1 minus the probability that it will occur. For example, if the probability of graduating within five years (P(A1)) is 0.47, the probability of not graduating (P(A2)) is 1 - 0.47 = 0.53.
• The **Law of Total Probability** provides a way to find probabilities of certain outcomes by considering all potential paths. It's used in the exercise to calculate \(P(W)\), or the probability of being female regardless of graduation, using given conditional probabilities (\(P(W | A_1)\) and \(P(W | A_2)\)).

These laws apply to a broad range of probability questions, not just those involving gender statistics or Bayes' theorem.

gender statistics

Gender statistics provide insight into patterns and probabilities related to gender-specific data. In the exercise, gender statistics come into play by analyzing how likely women and men are to graduate within five years. These statistics guide decision-making and policy development in educational contexts. Here are some key points about gender statistics in our problem:

• We assume women make up 50% of those graduating within five years, but only 45% of those who don't graduate in that timeframe. This shows a disparity that could warrant further investigation.
• Analyzing these statistics helps us compute the probability \(P(A_1|W)\), the probability of a female graduating within five years, which we found to be approximately 49.6%
• Similarly, calculating \(P(A_1|M)\), the probability of a male graduating in the same period, helps us compare the educational outcomes between genders.

By examining gender statistics through conditional probabilities and other probability laws, we gain a deeper understanding of educational success metrics by gender.