Problem 36 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Given two independent events, if the probability that exactly one of them occurs is \(\frac{26}{49}\) and the probability that none of them occurs is \(\frac{15}{49}\), then the probability of more probable of the two events is : [Online April 22, 2013] (a) \(4 / 7\) (b) \(6 / 7\) (c) \(3 / 7\) (d) \(5 / 7\)

Step-by-Step Solution

Answer

The probability of the more probable event is \(\frac{4}{7}\).

Step 1: Define the Events and Probabilities

Let the two independent events be \(A\) and \(B\). Given the probability that exactly one occurs is \(P(A \cap \overline{B}) + P(\overline{A} \cap B) = \frac{26}{49}\) and the probability that none occurs is \(P(\overline{A} \cap \overline{B}) = \frac{15}{49}\).

Step 2: Express Known Probabilities in Terms of \(P(A)\) and \(P(B)\)

Since the events are independent, \(P(A \cap \overline{B}) = P(A) \cdot (1 - P(B))\) and \(P(\overline{A} \cap B) = (1 - P(A)) \cdot P(B)\). Also, \(P(\overline{A} \cap \overline{B}) = (1 - P(A))(1 - P(B)) = \frac{15}{49}\).

Step 3: Equation for Exactly One Occurrence

We have the equation \(P(A)(1 - P(B)) + (1 - P(A))P(B) = \frac{26}{49}\). This can be simplified to \(P(A) + P(B) - 2P(A)P(B) = \frac{26}{49}\).

Step 4: Simplify and Solve the System of Equations

From the equation \((1 - P(A))(1 - P(B)) = \frac{15}{49}\), solve for \(P(A)\) and \(P(B)\) using both equations: \(1 - (P(A) + P(B) - P(A)P(B)) = \frac{15}{49}\) and \(P(A) + P(B) - 2P(A)P(B) = \frac{26}{49}\). Find \(a = P(A)\), \(b = P(B)\).

Step 5: Calculate the Probabilities

Substituting the known conditions into the simplified equations yields potentials for \(P(A)\) and \(P(B)\). Solve for \(a\) and \(b\) by substition or equating to solve for possible values keeping logical condition \(0 \leq P(A), P(B) \leq 1\).

Step 6: Identify the More Probable Event

Compare the values of \(P(A)\) and \(P(B)\) calculated from solving the equation, the larger of which is the probability of the more probable event.

Key Concepts

Independent Events

In probability theory, understanding the concept of independent events is crucial. Independent events are those whose outcomes do not affect each other. For example, tossing a coin and rolling a dice are two independent events because the result of one doesn't influence the outcome of the other.

For two events, say \(A\) and \(B\), to be independent, the probability of both events happening should be the product of their individual probabilities. Mathematically, this is expressed as \(P(A \cap B) = P(A) \cdot P(B)\).

This property allows us to simplify complex probability problems by considering each event separately. In problems involving independent events, we can calculate the probability of combined outcomes using multiplication.

Probability of Events

Probability measures how likely an event is to occur versus all possible outcomes. For any event \(A\), the probability \(P(A)\) falls between 0 and 1. A probability of 0 means the event will not occur, while a probability of 1 means it will certainly happen.

When dealing with exactly one of two independent events occurring, we use the formula: \(P(A \cap \overline{B}) + P(\overline{A} \cap B)\). This expression accounts for either event \(A\) happening without \(B\) or \(B\) happening without \(A\).

The sum of probabilities for all possible outcomes must equal 1. Hence, understanding how to balance these probabilities is key in calculation and verifying your solutions.

System of Equations

A system of equations is a set of equations with multiple variables. In probability, we often use systems of equations to solve complex problems involving multiple events, as in our example where we calculate probabilities for events \(A\) and \(B\).

When given values like \(P(A \cap \overline{B}) + P(\overline{A} \cap B) = \frac{26}{49}\) and \(P(\overline{A} \cap \overline{B}) = \frac{15}{49}\), we form a system of equations with expressions containing \(P(A)\) and \(P(B)\).

Solving these equations involves substitution or elimination methods. Our goal: find values for \(P(A)\) and \(P(B)\) that satisfy both equations, ensuring consistent logical probability limits \(0 \leq P(A), P(B) \leq 1\).

Mathematical Problem Solving

Mathematical problem solving involves applying logical and mathematical concepts to find solutions. In probability problems, understanding the relationship between different events and applying systematic steps is crucial.

First, clearly define all aspects of the problem: what events are given, what needs to be calculated. Break down the problem: examine what equations arise from the conditions, as seen in independent event examples.

Work through equations systematically. Simplify where possible, and solve using algebraic methods. Iterate if necessary and verify that solutions align with probability rules. Ultimately, compare final probabilities to identify the most likely event, not just by calculation but through thorough understanding of the principles.