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Chapter 4: Problem 90

The probability that a student graduating from Suburban State University has student loans to pay off after graduation is .60. The probability that a student graduating from this university has student loans to pay off after graduation and is a male is \(.24\). Find the conditional probability that a randomly selected student from this university is a male given that this student has student loans to pay off after graduation.

Short Answer

Expert verified
The conditional probability that a randomly selected student from this university is a male given that this student has student loans to pay off after graduation is \(0.24 / 0.60 = 0.4\) or 40%.

Step by step solution

01

Recognize the Given Probabilities

From the problem, we know that the probability that a student has a loan to pay off, denoted as P(L), is 0.60. Also, the probability that a student is a male and has loan to pay off, denoted as P(M 鈭� L), is 0.24.
02

Apply the Formula for Conditional Probability

We are asked to find the conditional probability that a student is a male given that this student has student loans to pay off after graduation. This can be denoted as P(M|L). According to the formula for conditional probability, P(M|L) = P(M 鈭� L) / P(L).
03

Calculate the Conditional Probability

Substitute the given probabilities into the formula: P(M|L) = P(M 鈭� L) / P(L) = 0.24 / 0.60. Perform the division to get the final answer.

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Key Concepts

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

Probability Theory
Probability Theory is a branch of mathematics that deals with the likelihood of events occurring. It forms the foundation for predicting future events based on known data.
This theory involves understanding different probability rules and formulas, which help in determining the chances of outcomes. In our case, we're dealing with conditional probability, which is essential when probability is influenced by a certain condition already being met.
For instance:
  • Unconditional Probability: This is the probability of an event occurring without any prior conditions, like the 0.60 probability of a student having loans.
  • Intersection of Events: The probability of two simultaneous events occurring is known as their intersection, or joint probability (P(M 鈭� L) = 0.24, in our exercise).
  • Conditional Probability: This calculates the likelihood of an event given another event has already happened. It's expressed as P(A|B), meaning the probability of A given B.
Understanding these fundamentals is vital to solve probability problems accurately.
Statistics
Statistics involves collecting, analyzing, interpreting, presenting, and organizing data, which allows us to make informed decisions based on numerical evidence.
Conditional probability is a prime example of statistical analysis. It involves deriving insights from given data about the relationship between different events. This method is particularly useful in fields like healthcare, finance, and social sciences.
In our exercise, gathering the data about students with loans and male students who have loans allows us to calculate the probability of these overlapping events.
  • Data Collection: Identifying relevant probabilities (0.60 and 0.24) forms the initial step in statistical analysis.
  • Data Analysis: Applying statistical techniques, such as using the formula for conditional probability, helps analyze this data efficiently.
Statistics provides the logical framework we need to learn and apply probability in real-life scenarios.
Bayes' Theorem
Bayes鈥� Theorem is a powerful part of probability theory that describes the probability of an event, based on prior knowledge of conditions that might be related to the event. This theorem provides a way to update our predictions or hypotheses when given new evidence.
Although not directly used in the original exercise, understanding Bayes' Theorem gives deeper insights into probabilistic inference. It combines conditional probability with a reverse perspective.
  • Bayesian Perspective: It emphasizes updating probabilities post-evidence. This is important when dealing with real-world uncertainty.
  • Formula: Bayes' Theorem is written as \( P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \), which is essential when needing to switch aspect views of probability, such as predicting causes after observing effects.
Bayes鈥� Theorem is instrumental in fields like machine learning, where algorithms make predictions based on evolving data.

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Most popular questions from this chapter

A random sample of 250 juniors majoring in psychology or communication at a large university is selected. These students are asked whether or not they are happy with their majors. The following table gives the results of the survey. Assume that none of these 250 students is majoring in both areas. $$ \begin{array}{lcc} \hline & \text { Happy } & \text { Unhappy } \\ \hline \text { Psychology } & 80 & 20 \\ \text { Communication } & 115 & 35 \\ \hline \end{array} $$ a. If one student is selected at random from this group, find the probability that this student is i. happy with the choice of major ii. a psychology major iii. a communication major given that the student is happy with the choice of major iv. unhappy with the choice of major given that the student is a psychology major v. a psychology major and is happy with that major vi. a communication major \(o r\) is unhappy with his or her major b. Are the events "psychology major" and "happy with major" independent? Are they mutually exclusive? Explain why or why not.

Five percent of all items sold by a mail-order company are returned by customers for a refund. Find the probability that of two items sold during a given hour by this company, a. both will be returned for a refund b. neither will be returned for a refund Draw a tree diagram for this problem.

Given that \(P(B \mid A)=.70\) and \(P(A\) and \(B)=.35\), find \(P(A)\).

Refer to Exercise 4.48. A 2010-2011 poll conducted by Gallup (www.gallup.com/poll/148994/ Emotional-Health-Higher-Among-Older- Americans.aspx) examined the emotional health of a large number of Americans. Among other things, Gallup reported on whether people had Emotional Health Index scores of 90 or higher, which would classify them as being emotionally well-off. The report was based on a survey of 65,528 people in the age group \(35-44\) years and 91,802 people in the age group \(65-74\) years. The following table gives the results of the survey, converting percentages to frequencies. $$ \begin{array}{lcc} \hline & \text { Emotionally Well-Off } & \text { Emotionally Not Well-Off } \\\ \hline \text { 35-44 Age group } & 16,016 & 49,512 \\ \text { 65-74 Age group } & 32,583 & 59,219 \\ \hline \end{array} $$ a. Suppose that one person is selected at random from this sample of 157,330 Americans. Find the following probabilities. i. \(P(35-44\) age group and emotionally not well-off \()\) ii. \(P(\) emotionally well-off and \(65-74\) age group \()\) b. Find the joint probability of the events \(35-44\) age group and \(65-74\) age group. Is this probability zero? Explain why or why not.

Five hundred employees were selected from a city's large private companies and asked whether or not they have any retirement benefits provided by their companies. Based on this information, the following two-way classification table was prepared. $$ \begin{array}{lcc} &{\text { Have Retirement Benefits }} \\ \hline { 2 - 3 } & \text { Yes } & \text { No } \\ \hline \text { Men } & 225 & 75 \\ \text { Women } & 150 & 50 \\ \hline \end{array} $$ a. Suppose one employee is selected at random from these 500 employees. Find the following probabilities. i. Probability of the intersection of events "woman" and "yes" ii. Probability of the intersection of events "no" and "man" b. Mention what other joint probabilities you can calculate for this table and then find them. You may draw a tree diagram to find these probabilities.
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