Problem 8 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

In a college \(25 \%\) students fail in maths, \(15 \%\) fail in chemistry and \(10 \%\) students fail in maths and chemistry both. A student is selected at random, then (i) What is the probability that he fails in Maths, if he is failed in Chemistry? (ii) What is the probability that he fails in Chemistry, if he is failed in Maths? (iii) What is the probability that he is failed in Maths or Chemistry?

Step-by-Step Solution

Answer

(i) Probability is \( \frac{2}{3} \). (ii) Probability is \( \frac{2}{5} \). (iii) Probability is \( 0.30 \).

Step 1: Understanding Given Probabilities

We are given that 25% of students fail in maths, which means the probability that a student fails in maths is \( P(M) = 0.25 \). Similarly, 15% fail in chemistry, so \( P(C) = 0.15 \). Additionally, 10% fail in both subjects, thus \( P(M \cap C) = 0.10 \).

Step 2: Finding Probability of Failing Maths if Failed in Chemistry

We apply conditional probability for part (i): \( P(M|C) \). This is calculated using the formula for conditional probability: \( P(M|C) = \frac{P(M \cap C)}{P(C)} \). Substituting the known values, we get \( P(M|C) = \frac{0.10}{0.15} = \frac{2}{3} \).

Step 3: Finding Probability of Failing Chemistry if Failed in Maths

For part (ii), we find the conditional probability \( P(C|M) \). Using the formula for conditional probability, \( P(C|M) = \frac{P(M \cap C)}{P(M)} \). Substituting the given probabilities, \( P(C|M) = \frac{0.10}{0.25} = \frac{2}{5} \).

Step 4: Finding Probability of Failing Either Maths or Chemistry

For part (iii), we use the formula \( P(M \cup C) = P(M) + P(C) - P(M \cap C) \). Plug in the values: \( P(M \cup C) = 0.25 + 0.15 - 0.10 = 0.30 \). This is the probability that a student fails in either maths or chemistry.

Key Concepts

Probability Theory

Probability Theory is a branch of mathematics that deals with calculating the likelihood of various outcomes in random events. It provides a way to predict and quantify the uncertainty of events occurring. In our exercise, we are interested in understanding the chances of a student failing specific subjects – maths and chemistry. The concept of conditional probability is critical here, as it allows us to find the probability of an event occurring, given that another event has already occurred.

When we talk about conditional probability, we use notation like \( P(A|B) \), which denotes the probability of event A happening given that event B has occurred. This is a foundational concept because real-world situations often require us to calculate probabilities based on known conditions.

• The formula for conditional probability is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), indicating that you divide the probability of both events happening by the probability of the known condition.

This method is applied in our solution steps, like finding the conditional probabilities \( P(M|C) \) and \( P(C|M) \), related to students failing in subjects based on given conditions. This makes the problem-solving concise and logical, fitting the practical aspects of Probability Theory.

Mathematics Education

Mathematics Education encompasses teaching and learning practices, aiming to make complex topics accessible to students. One crucial aspect is simplifying abstract concepts like conditional probability in an educational context. In the original exercise, understanding how conditions affect outcomes in probability illustrates a typical real-world application, aiding students in grasping the relevance of mathematics.

Effective math education often involves:
- Clear explanations and logical progression of ideas.
- Using relatable examples and situations.
- Breaking down complex problems into manageable parts.

Our exercise does just that by guiding students through each step, ensuring they learn how to interpret conditions and apply formulas accurately. Such structured learning experiences are vital because they help build foundational skills in probability, which are essential for advanced studies.

Algebra 2 Concepts

Algebra 2 builds on foundational algebraic ideas, introducing advanced concepts such as probability, functions, and complex numbers. The exercise leverages Algebra 2 skills by utilizing knowledge of set operations and understanding intersections, unions, and complements. These are vital for solving problems related to probability.

In probabilistic terms, we use:

• Intersection (\( P(M \cap C) \)): The probability that both events M and C happen.
• Union (\( P(M \cup C) \)): The probability that either event M or C happens, calculated using the formula \( P(M) + P(C) - P(M \cap C) \).

The exercise demonstrates the application of these concepts in practical scenarios, showing students how algebraic reasoning and set operations translate the question into a solvable mathematical model. This kind of problem-solving framework is central to Algebra 2, equipping students with the skills needed to tackle higher-level mathematics and understand statistical data.