Problem 6 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(P(A)=\frac{1}{2}, P(B)=\frac{1}{4}\) and \(P(A \cap B)=\frac{1}{4}\), then find the value of the following: (i) \(P\left(\frac{A}{B}\right)\) (ii) \(P\left(\frac{B}{A}\right)\) (iii) \(P(A \cup B)\)

Step-by-Step Solution

Answer

(i) \( P(A|B) = 1 \), (ii) \( P(B|A) = \frac{1}{2} \), (iii) \( P(A \cup B) = \frac{1}{2} \).

Step 1: Solve for Conditional Probability P(A|B)

To find the conditional probability of event A given event B, use the formula:\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]Here, \( P(A \cap B) = \frac{1}{4} \) and \( P(B) = \frac{1}{4} \). Substitute these values into the formula:\[ P(A|B) = \frac{\frac{1}{4}}{\frac{1}{4}} = 1 \]

Step 2: Solve for Conditional Probability P(B|A)

To find the conditional probability of event B given event A, use the formula:\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]Given \( P(A \cap B) = \frac{1}{4} \) and \( P(A) = \frac{1}{2} \), substitute these values:\[ P(B|A) = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2} \]

Step 3: Solve for Probability P(A ∪ B)

To find the probability of the union of events A and B, use the formula:\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]Substitute the given probabilities \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{4} \), and \( P(A \cap B) = \frac{1}{4} \):\[ P(A \cup B) = \frac{1}{2} + \frac{1}{4} - \frac{1}{4} = \frac{1}{2} \]

Key Concepts

Understanding Conditional Probability

Conditional probability is a fascinating concept that allows us to understand how the probability of an event changes when we have additional information. In simple terms, it answers the question: "What is the probability of an event A happening if we know that another event B has already happened?"

The mathematical formula for conditional probability is expressed as \[P(A|B) = \frac{P(A \cap B)}{P(B)}\].

This formula tells us that the probability of event A given B is the probability of both events occurring together divided by the probability of event B. In most problems, this formula will help you find the likelihood of one event based on the occurrence of another.

Here's a step-by-step guide on using the formula:

• Identify the events you're interested in (A and B).
• Calculate or use given values for \(P(A \cap B)\)—the probability of both A and B happening.
• Determine \(P(B)\)—the probability of B, which acts as the total scope for our scenario.
• Substitute these values into the formula to find \(P(A|B)\).

This step helps in scenarios where knowing a specific piece of information (B) provides new insights into the likelihood of event A.

The Union of Events in Probability

In probability theory, the union of events is a concept used to calculate the likelihood of at least one of a set of events happening. The mathematical representation for the union is \(P(A \cup B)\), which means "the probability of either A or B, or both, occurring".

The formula to find the union of two events is:\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\].

Why do we subtract \(P(A \cap B)\)? It's because the probabilities of A and B include the overlap (occurrence of both), so we remove the shared part to avoid double counting. Here's how to use the formula:

• Find \(P(A)\), the probability of the first event happening.
• Find \(P(B)\), the probability of the second event occurring.
• Calculate \(P(A \cap B)\)—the probability that both events occur together.
• Use these values in the formula to understand the combined likelihood.

The union of events is particularly useful when evaluating the probability of multiple outcomes that can happen simultaneously or exclusively.

The Intersection of Events Explained

The intersection of events concerns scenarios where two events happen at the same time. This is represented by \(P(A \cap B)\), which reads "the probability that both events A and B will occur".

This concept is crucial when understanding dependencies between events. In cases where events are independent, the intersection can be calculated simply by multiplying their probabilities: \[P(A \cap B) = P(A) \times P(B)\].

However, when they aren't independent, more information (like conditional probabilities) might be needed. The intersection directly links to conditional probability with the same formula rearrangement used earlier: \(P(A \cap B) = P(B) \times P(A|B)\).

Here's how to think of intersections:

• If both events A and B happening means combined restriction, think intersection.
• Know that this helps in understanding overlaps between event probabilities.
• Use intersections to assess the extent to which two events relate in a shared scenario.

In practice, understanding intersections can clarify how two distinct conditions might impact each other when dealing with probability calculations.