Problem 42 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

It is given that the events \(A\) and \(B\) are such that \(P(A)=\frac{1}{4}, P(A \mid B)=\frac{1}{2}\) and \(P(B \mid A)=\frac{2}{3} .\) Then \(P(B)\) is [2008] (a) \(\frac{1}{6}\) (b) \(\frac{1}{3}\) (c) \(\frac{2}{3}\) (d) \(\frac{1}{2}\)

Step-by-Step Solution

Answer

\(P(B) = \frac{1}{3}\), option (b).

Step 1: Understanding Given Probabilities

We are provided with three probabilities: \(P(A) = \frac{1}{4}\), \(P(A \mid B) = \frac{1}{2}\), and \(P(B \mid A) = \frac{2}{3}\). These are known as conditional probabilities and direct probabilities.

Step 2: Use Bayes' Theorem

To find \(P(B)\), start with Bayes' Theorem: \(P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}\). We will rearrange this formula later to solve for \(P(B)\).

Step 3: Rearrange Bayes' Theorem

Rearrange the formula from Step 2 to express \(P(B)\) in terms of the known probabilities.\[ P(B) = \frac{P(B \mid A) \cdot P(A)}{P(A \mid B)} \]

Step 4: Substitute Given Values

Substitute the known values into the rearranged formula:\[ P(B) = \frac{\frac{2}{3} \times \frac{1}{4}}{\frac{1}{2}} \]

Step 5: Perform the Calculations

First, calculate the numerator: \(\frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}\).Next, divide by the denominator \(\frac{1}{2}\):\[ P(B) = \frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{6} \times 2 = \frac{1}{3} \]

Step 6: Consolidate the Solution

Thus, the probability \(P(B)\) is \(\frac{1}{3}\).

Key Concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory that describes the likelihood of an event occurring given that another event has already occurred. It's all about using the additional information to update the probability of an event.
For example, in the exercise, we have the conditional probabilities \(P(A \mid B) = \frac{1}{2}\) and \(P(B \mid A) = \frac{2}{3}\). Here, \(P(A \mid B)\) means "the probability of event \(A\) occurring given that \(B\) has already occurred."
To calculate any conditional probability, use the formula:
\[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\]where \(P(A \cap B)\) is the probability that both \(A\) and \(B\) happen simultaneously. Understanding this concept helps in making more informed probability calculations when dealing with dependent events.

Probability Calculation

Probability calculation is at the heart of solving probability problems. It involves using known values and mathematical formulas to find unknown probabilities. In the given exercise, we applied probability calculations in several steps.
First, we identified that we could use Bayes' Theorem to find the missing probability \(P(B)\). This is a common approach when dealing with problems involving conditional probabilities. The relevant formula from Bayes' Theorem is:
\[ P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)} \] By rearranging, it becomes:
\[ P(B) = \frac{P(B \mid A) \cdot P(A)}{P(A \mid B)} \]
Substituting the known values into this equation allows us to compute \(P(B)\) using straightforward arithmetic operations. This step-by-step calculation illustrates how systematic probability calculations help us derive unknown probabilities from known pieces of information.

Probability Theory

Probability theory provides the framework and rules for dealing with random events and their likelihoods. It allows us to quantify how likely events are to happen and make predictions based on mathematical principles. This theory encompasses various concepts such as events, outcomes, and probability measures.
In the exercise, probability theory helps us understand how probabilities are interrelated, such as how the probability of \(B\) depends on the given probabilities of \(A\) and the conditional probabilities \(P(A \mid B)\) and \(P(B \mid A)\).

• **Events**: Basic outcomes like \(A\) and \(B\), with each having specified probabilities.
• **Probability Measures**: Quantifying the chance of an event's occurrence, like \(P(A) = \frac{1}{4}\).
• **Bayes' Theorem**: A pivotal rule in probability theory that relates conditional probabilities and offers a method to compute unknown probabilities.

Understanding probability theory is crucial for interpreting and solving complex probability problems, ensuring predictions are accurate and formulas are correctly applied.