Problem 101 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(A, B, C\) be three events. If the probability of occurring exactly one event out of \(A\) and \(B\) is \(1-a\), out of \(B\) and \(C\) is \(1-2 a\), out of \(C\) and \(A\) is \(1-a\) and that of occurring three events simultaneously is \(a^{2}\), then the probability that at least one out of \(A, B, C\) will occur, is \((\mathrm{A})<\frac{1}{2}\) (B) \(>\frac{1}{3}\) \((\mathrm{C})>\frac{1}{2}\) (D) \(<\frac{1}{3}\)

Step-by-Step Solution

Answer

C: >\frac{1}{2}

Step 1: Write Down Given Probabilities

We know the following probabilities: \( P(A \cup B) - P(A \cap B) = 1-a \), \( P(B \cup C) - P(B \cap C) = 1-2a \), \( P(C \cup A) - P(C \cap A) = 1-a \), and \( P(A \cap B \cap C) = a^2 \).

Step 2: Reformulating Known Probabilities

Express the given terms: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), \( P(B \cup C) = P(B) + P(C) - P(B \cap C) \), \( P(C \cup A) = P(C) + P(A) - P(C \cap A) \).

Step 3: Use Formulas to Find Intersection Terms

Parse each event combination: \( P(A \cap B) = P(A) + P(B) - 1 + a \), \( P(B \cap C) = P(B) + P(C) - 1 + 2a \), \( P(C \cap A) = P(C) + P(A) - 1 + a \).

Step 4: Solve for Total Probability of Either Event

Subtract from the total probability including intersections: \( P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C) \). Substituting the known variables will simplify the equation.

Step 5: Substitute and Simplify

With substitution, this becomes: \[ P(A \cup B \cup C) = 3(P(A) + P(B) + P(C)) - 2(1-a) - (1-2a) + a^2 \], resulting in aligning terms and simplifying expressions.

Step 6: Calculate Using Dependencies Among Variables

Simplify further using assumptions to express dependencies among terms and independent solutions as needed to ensure right satisfaction of conditions.

Step 7: Evaluate Probability Expression

Final step is achieving simplified form from a long expression, solve or re-arrange depending on algebraic residuals to fit provided criteria.

Key Concepts

Events and Sets

In probability theory, an event is a set of outcomes of an experiment. For example, when you toss a coin, the possible outcomes are "heads" or "tails." In this case, an event can be defined as obtaining a head or a tail ensued by the coin toss.
A **set**, in mathematics, is a collection of distinct objects. Sets are often used to define probability events because they provide a structured way to arrange possible outcomes.
For instance, if we have three events, say \(A\), \(B\), and \(C\), each could represent an occurrence such as drawing a red ball from a bag for \(A\), a blue ball for \(B\), and a yellow ball for \(C\). If sets allow us to list these specific outcomes, events provide the framework within which these outcomes are measured in terms of probability.

Probability Formulas

Probability formulas are essential tools for calculating the likelihood of events.

• **Union of two events \(A\) and \(B\):** The probability that event \(A\) or \(B\) occurs is denoted as \(P(A \cup B)\) and is calculated by the formula \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

This accounts for the overlap where both \(A\) and \(B\) occur simultaneously, represented as \(P(A \cap B).\)

• **Intersection of events:** It refers to events happening together. Here, \(P(A \cap B)\) equals the probability of both events \(A\) and \(B\) occurring.
• **Probability of complementary events:** If you know the probability of an event \(A\), then the probability of \(A\) not occurring is \(1 - P(A).\)

These formulas become the building blocks as you solve complex problems like determining the probability of at least one event occurring out of \(A, B, C\).

Union and Intersection of Events

Understanding the concepts of union and intersection of events is crucial in probability.

• **Union (\(\cup\)):** When we talk about the union of events \(A\) and \(B\), denoted as \(A \cup B\), we refer to the occurrence of either event \(A\), event \(B\), or both. It encapsulates all possible outcomes in either event.
• **Intersection (\(\cap\)):** Conversely, the intersection \(A \cap B\) considers only those outcomes that are common to both events \(A\) and \(B\). It represents simultaneous occurrences.

In the exercise, to find the probability that at least one of the events \(A\), \(B\), or \(C\) occurs, we calculate the union \(P(A \cup B \cup C)\). This involves taking into account multiple intersections and ensuring no probabilities are double-counted.
The formula we use is:\[P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C)\]This ensures a comprehensive account of all possible outcomes of the events when at least one occurs.