Problem 30 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Two events \(A\) and \(B\) are such that \(P(A)=1 / 4\) \(P(B / A)=1 / 2, P(A / B)=1 / 4\), then \(P(\bar{A} / \bar{B})\) is equal to (a) \(1 / 4\) (b) \(3 / 4\) (c) \(1 / 2\) (d) \(2 / 3\)

Step-by-Step Solution

Answer

The answer is (b) \(\frac{3}{4}\).

Step 1: Understanding Conditional Probability

Given that \(P(A) = \frac{1}{4}\), \(P(B|A) = \frac{1}{2}\), and \(P(A|B) = \frac{1}{4}\). We need to find \(P(\bar{A} | \bar{B})\). Here, \(\bar{A}\) and \(\bar{B}\) represent the complements of events \(A\) and \(B\), respectively.

Step 2: Using the Formula for Conditional Probability

Conditional probability \(P(A|B)\) and \(P(B|A)\) can be represented using the formulas \(P(A|B) = \frac{P(A \cap B)}{P(B)}\) and \(P(B|A) = \frac{P(A \cap B)}{P(A)}\).

Step 3: Solving for \(P(B)\) Using \(P(A|B)\)

We have \(P(A|B) = \frac{1}{4}\). Therefore, \(\frac{P(A \cap B)}{P(B)} = \frac{1}{4}\). Rearranging, we find \(P(A \cap B) = \frac{1}{4}P(B)\).

Step 4: Solving for \(P(A \cap B)\) Using \(P(B|A)\)

Given \(P(B|A) = \frac{1}{2}\), we have \(\frac{P(A \cap B)}{\frac{1}{4}} = \frac{1}{2}\). Solving, \(P(A \cap B) = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}\).

Step 5: Equating Two Expressions for \(P(A \cap B)\)

From Steps 3 and 4, \(\frac{1}{4}P(B) = \frac{1}{8}\). Solving for \(P(B)\), we find \(P(B) = \frac{1}{2}\).

Step 6: Using Complementary Probability Relations

Now, \(P(A \cap \bar{B}) = \frac{1}{4} - \frac{1}{8} = \frac{1}{8}\) because \(P(A) = \frac{1}{4}\) and \(P(A \cap B) = \frac{1}{8}\). Also, \(P(\bar{B}) = 1 - P(B) = 1 - \frac{1}{2} = \frac{1}{2}\).

Step 7: Calculating \(P(\bar{A} | \bar{B})\)

Finally, using \(P(\bar{A} | \bar{B}) = \frac{P(\bar{A} \cap \bar{B})}{P(\bar{B})}\). Note that \(P(\bar{A} \cap \bar{B}) = P(\bar{B}) - P(A \cap \bar{B}) = \frac{1}{2} - \frac{1}{8} = \frac{3}{8}\). Thus, \(P(\bar{A} | \bar{B}) = \frac{3/8}{1/2} = \frac{3}{4}\).

Key Concepts

Complementary Events

In probability, understanding complementary events can simplify the calculation of probabilities. A complementary event refers to the notion that if one event occurs, the other does not. For an event \(A\), the complement, \(\bar{A}\), represents all outcomes that are not in \(A\). Here's how it works:

• If the probability of event \(A\) happening is \(P(A)\), then the complement \(\bar{A}\) has a probability of \(P(\bar{A}) = 1 - P(A)\).
• This relationship means if we know the probability of an event happening, we can easily find the likelihood of it not happening.

When dealing with two events like \(A\) and \(B\), their complementarities play a crucial role. For example, finding \(P(\bar{A} \cap \bar{B})\) involves understanding both the events not happening together. Like in our problem, where calculating \(P(\bar{A} | \bar{B})\) depends on first determining the joint complement \(\bar{A} \cap \bar{B}\). This inherently makes the concept of complementary events a foundational piece in probability calculations, especially in more complex scenarios.

Probability Formulas

Probability formulas provide the mathematical foundation for understanding relationships between events. Two key formulas for conditional probability are:

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

These formulas help relate the probability of \(A\) given \(B\) has occurred to the joint probability \(P(A \cap B)\). This was crucial in our exercise where we used both formulas to calculate \(P(\bar{A} | \bar{B})\).
Calculating probabilities using these formulas involves following steps:

• Identify the known probabilities \(P(A)\), \(P(B|A)\), or similar given information.
• Use rearranging steps to find missing joint probabilities like \(P(A \cap B)\).
• Solve for unknowns such as \(P(B)\) using equations derived from these formulas.

This helpfully clarifies relationships between different events and their outcomes. Understanding these probability formulas allows for precise calculations, particularly in complex scenarios involving conditional probability.

Conditional Probability Calculation

Conditional probability calculation is about finding the likelihood of an event occurring, given that another event has already happened. It helps us focus more narrowly on the probability landscape. In our example, calculating \(P(\bar{A} | \bar{B})\) required several steps and known probabilities:

• Initially, we determined \(P(A \cap B)\) using conditional probability \(P(B|A)\).
• The joint probability was then equalized with the expression from \(P(A|B)\) to solve for \(P(B)\).
• Using complementary events, we calculated \(P(\bar{B}) = 1 - P(B)\).
• Subsequently, \(P(\bar{A} | \bar{B})\) was found using the equation \(\frac{P(\bar{A} \cap \bar{B})}{P(\bar{B})}\).

This demonstrates that conditional probability involves a mix of employing conditional formulas, joint probabilities, and understanding complementary relationships. The result gives us insightful precision in evaluating the nuanced outcomes in various statistical situations. By considering these steps and calculations, you can effectively solve and understand conditional probability problems.