Problem 25 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If two events \(A\) and \(B\) are such that \(P(\bar{A})=0.3, P(B)=0.4\) and \(P(A \bar{B})=0.5\) then \(P\left(\frac{B}{A \cup \bar{B}}\right)\) equals (a) \(1 / 2\) (b) \(1 / 4\) (c) \(1 / 3\) (d) None of these

Step-by-Step Solution

Answer

The correct answer is (b) \(\frac{1}{4}\).

Step 1: Understand the Problem

We are given probabilities of events and need to calculate a conditional probability. We have \(P(\bar{A}) = 0.3\), \(P(B) = 0.4\), and \(P(A \bar{B}) = 0.5\). We need to find \(P(B | A \cup \bar{B})\).

Step 2: Identify Relevant Formulas

Conditional probability \(P(B|C)\) is calculated by \(P(B \cap C) / P(C)\). For this exercise, we need to find \(P(B \cap (A \cup \bar{B}))\) then divide by \(P(A \cup \bar{B})\).

Step 3: Calculate Basic Probabilities

We are given \(P(\bar{A}) = 0.3\), so \(P(A) = 1 - P(\bar{A}) = 0.7\). We have \(P(A \bar{B}) = 0.5\), which implies \(P(A \cap B) = P(A) - P(A \bar{B}) = 0.7 - 0.5 = 0.2\).

Step 4: Find Probability of Intersection

\(P(B \cap (A \cup \bar{B}))\) is equivalent to \(P((B \cap A) \cup (B \cap \bar{B}))\) which simplifies to \(P(B \cap A) = 0.2\) since \((B \cap \bar{B})\) is an empty set.

Step 5: Calculate Union Probability

Now calculate \(P(A \cup \bar{B})\) using \(P(A \cup \bar{B}) = P(A) + P(\bar{B}) - P(A \cap \bar{B})\). \(P(\bar{B}) = 1 - P(B) = 0.6\), and thus \(P(A \cup \bar{B}) = 0.7 + 0.6 - 0.5 = 0.8\).

Step 6: Compute Conditional Probability

Now find \(P(B | A \cup \bar{B}) = \frac{P(B \cap (A \cup \bar{B}))}{P(A \cup \bar{B})} = \frac{0.2}{0.8} = 0.25\).

Step 7: Determine the Correct Option

The value obtained is \(0.25\), which is equivalent to \(\frac{1}{4}\). Hence, the correct option is (b) \(\frac{1}{4}\).

Key Concepts

Probability Theory

Probability theory is the branch of mathematics concerning the analysis and calculation of likelihoods or chances of different outcomes. It's used to quantify the concept of uncertainty. In simpler terms, probability answers the question: "How likely is it that a given event will happen?". Here's how probability is usually expressed:

• The probability of any event, denoted as \(P(E)\), is a number between 0 and 1.
• A probability of 0 means the event will not occur, while 1 means it will certainly occur.
• For example, the probability of getting a head on a balanced coin toss is \(P(\text{Head}) = 0.5\).

This exercise involves finding a conditional probability, which represents the probability of an event happening given another event has occurred. For events \(A\) and \(B\), it's written as \(P(B|A)\), and calculated as:\[P(B|A) = \frac{P(A \cap B)}{P(A)}\]Using this formula helps us determine the likelihood of \(B\) given that \(A\) has occurred.

Set Operations in Probability

Set operations play a crucial role in probability as they help in finding the probabilities of combined events. The most common set operations in probability are union, intersection, and complement.

• Union (\(A \cup B\)): This operation combines events \(A\) and \(B\) to include all outcomes that are in either of the events. The probability of the union is the likelihood that either \(A\) or \(B\) or both occur.
• Intersection (\(A \cap B\)): This is the set of outcomes that \(A\) and \(B\) have in common. The probability of the intersection is the likelihood that both \(A\) and \(B\) occur at the same time.
• Complement (\(\bar{A}\)): This gives the probability that event \(A\) does not happen. It's expressed as \(1 - P(A)\).

In solving problems like the given exercise, these operations help break down complex event relationships into simpler, more manageable parts by focusing on how events interact with each other.

Union and Intersection of Events

Understanding the union and intersection of events is essential in probability calculations. This concept involves knowing how different events overlap or combine, and affects how we calculate overall probabilities.Let's explore each:

• Union \((A \cup B)\): This operation refers to all the outcomes that are part of either event \(A\), event \(B\), or both. If we are interested in the probability of \(A\) or \(B\) happening, we deal with their union.
• Intersection \((A \cap B)\): This involves only the outcomes that both events share, where both occur simultaneously. If a task requires knowing the chance of both events happening, we use intersection.

The exercise provides an opportunity to apply these concepts in practice, using them to find conditional probabilities. Through set operations, we derived critical steps like calculating \(P(B \cap (A \cup \bar{B}))\), turning complex formulas into simpler calculations. By mastering these fundamental aspects, one is better equipped to decipher and solve intricate probabilistic scenarios.