Problem 126 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The probability that a student passes in Mathematics, Physics and Chemistry are \(m, p\) and \(c\), respectively. Of these subjects, the student has a \(75 \%\) chance of passing in at least one, a \(50 \%\) chance of passing in at least two and a \(40 \%\) chance of passing in exactly two. Which of the following relations are true? (A) \(p+m+c=19 / 20\) (B) \(p+m+c=27 / 20\) (C) \(p m c=1 / 10\) (D) \(p m c=1 / 4\)

Step-by-Step Solution

Answer

The correct relations are: (A) and (C) are true.

Step 1: Understanding Overlapping Probabilities

To solve this problem, we recognize the settings of overlapping probabilities. Let \( P(A) \), \( P(B) \), and \( P(C) \) be the probabilities of the student passing in mathematics, physics, and chemistry, respectively, represented by \( m \), \( p \), and \( c \). We need to explore the probability of passing one or more, two or more, and exactly two subjects.

Step 2: Calculate Probability of Passing At Least One

We use the given probability of passing at least one subject, \( P(A \cup B \cup C) = 0.75 \). The formula for the union of three sets is:\[P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C). \]Given \( P(A \cup B \cup C) = 0.75 \), we use this in conjunction with other available probabilities.

Step 3: Calculate Probability of Passing At Least Two

From the problem, passing at least two subjects has a probability \( P((A \cap B) \cup (B \cap C) \cup (C \cap A)) = 0.50 \). This is represented by:\[P((A \cap B) \cup (B \cap C) \cup (C \cap A)) = P(A \cap B) + P(B \cap C) + P(C \cap A) - 2P(A \cap B \cap C) = 0.50\]

Step 4: Calculate Probability of Passing Exactly Two

The probability of passing exactly two subjects is given as \( P( ext{exactly two}) = 0.4 \). This can be formulated as:\[P(A \cap B \cap \overline{C}) + P(A \cap \overline{B} \cap C) + P(\overline{A} \cap B \cap C) = 0.40\]

Step 5: Solving for Individual Probabilities

By combining the provided probabilities to create equations, solve them step-by-step. Utilize the formulas in prior steps to deduce logical equations

Key Concepts

Overlapping Probabilities

In probability theory, overlapping probabilities deal with the chance of multiple events occurring at the same time. When several events can happen simultaneously, their probabilities may overlap. This is represented using mathematical formulas that take into account both individual and joint probabilities of these events.

In our example problem, the events are passing various subjects: Mathematics, Physics, and Chemistry. To find the probability of passing at least one subject, we use the formula for the probability of the union of three events:

• \[ P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C). \]

This formula calculates the probability that a student passes one or more subjects by adding the individual probabilities and subtracting overlapping probabilities to adjust for any overcounted intersections.

Understanding these calculations helps us maintain accuracy in combined probabilistic events. It clears confusion about how overlapping areas contribute to the total probability.

Conditional Probability

The concept of conditional probability is an essential part of probability theory. It refers to the likelihood of an event occurring given that another event has already occurred. Conditional probability answers questions such as "Given that the student passes Mathematics, what is the probability they also pass Physics?".

This is typically denoted as \(P(A|B)\), meaning the probability of event \(A\) happening when \(B\) is already true. It's calculated using the formula:

• \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

In the specific problem we are working with, although not directly given, conditional probabilities can help us better understand how passing in one subject might affect the probabilities of passing in others. By considering relationships between probabilities of passing two subjects, like \( P(A \cap B) \), the structure of conditional probability allows deeper insight into these connections.

Probability Theory

Probability theory is the mathematical foundation upon which probability concepts are built. It provides tools and frameworks to model uncertain situations and evaluate the likelihood of different outcomes. At its core, probability theory deals with the analysis of random events and the laws that govern them.

In our example, probability theory helps us quantify the chances of a student passing different subjects. By using concepts such as the addition rule for probabilities and understanding events' intersections, we can find coherent solutions to complex problems involving multiple events.

A strong grasp of probability theory is crucial, as it plays an indispensable role in fields ranging from statistics, insurance, gaming, all the way to science and everyday decision-making. The foundations of probability theory, like any precise scientific system, offer both predictive power and explanatory understanding over random processes.