Let A and B be two events such that the probability that exactly one of them occurs is 2/5 and the p

Q: Wait, why can't we just use the formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$?

That formula is absolutely correct, but it requires knowing $P(A)$ and $P(B)$ individually. In this problem, we don't have those values. Instead, we are given the aggregate probability of the union and the aggregate probability of the 'exactly one' region. By focusing on the disjoint regions of the Venn diagram, we bypass the need for individual probabilities entirely.

Q: What does 'exactly one' mean mathematically?

It refers to the symmetric difference of the two sets, denoted as $(A \setminus B) \cup (B \setminus A)$. It represents all outcomes that are in $A$ or $B$, but not in both. In a Venn diagram, this is the area of the two circles excluding the central overlap.

Q: Is it possible for the probability of the intersection to be zero?

Yes, if the events are mutually exclusive, the intersection is empty, and its probability is zero. In this problem, the math dictates the value, but always remember that $P(A \cap B)$ must be a value between $0$ and $1$.

Question

Let $A$ and $B$ be two events such that the probability that exactly one of them occurs is $2/5$ and the probability that $A$ or $B$ occurs is $1/2$, then the probability of both of them occur together is

Options:

(A) 0.10

(B) 0.20

(C) 0.01

(D) 0.02

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a whiteboard, and you draw two circles, $A$ and $B$, overlapping in the center. This is your Venn diagram, your map to the territory of uncertainty.

The entire shaded area covered by both circles is the union, $P(A \cup B)$, which represents the probability that at least one of these events occurs. We are given that this value is:

$$P(A \cup B) = \frac{1}{2}$$

Now, consider the 'exactly one' region. This is the part of the circles that excludes the central overlap—the 'crescent moons' of our diagram. We are given this probability as:

$$P(	ext{Exactly one}) = \frac{2}{5}$$

The objective is to find the probability of both events occurring together, which is the intersection, $P(A \cap B)$.

### The Logical Bridge

The union $P(A \cup B)$ is simply the sum of the 'exactly one' region and the intersection region. Think of it as a puzzle.

If you take the total area of the union and subtract the 'exactly one' area, you are left with the central overlap, the intersection. The equation is beautifully simple:

$$P(A \cap B) = P(A \cup B) - P(	ext{Exactly one})$$

This is the logical bridge that connects our given values to the final result.

### The Final Calculation

Now, let us perform the arithmetic with precision. Substituting our known values into the equation:

$$P(A \cap B) = \frac{1}{2} - \frac{2}{5}$$

To subtract these, we need a common denominator. The least common multiple of $2$ and $5$ is $10$. We convert the fractions accordingly:

$$\frac{1}{2} = \frac{5}{10} \quad 	ext{and} \quad \frac{2}{5} = \frac{4}{10}$$

Performing the subtraction:

$$\frac{5}{10} - \frac{4}{10} = \frac{1}{10}$$

Converting this fraction to a decimal, we obtain the final result:

**$P(A \cap B) = 0.1$**

By visualizing the geometry of the sets, we have turned a potentially confusing problem into a clear, logical path. Remember, in JEE Advanced, the most complex problems often yield to the simplest visual models.

Animated Solution for Mathematics - Probability: Let $A$ and $B$ be two events such that the probability that exactly one of them occurs is $2/5$ and the probability that $A$ or $B$ occurs is $1/2$ , then the probability of both of them occur together is

Select Answer:

Visualized Solution

Introduction to Events $A$ and $B$

Visualizing the Union $P (A \cup B)$

Defining 'Exactly One' Event

The Logic Bridge: $P (A \cap B)$

Raw Setup (Substitution)

Finding the Common Denominator

Atomic Calculation: $P (A \cap B)$

Final Result in Decimal

The Sigma Insight: Addition and Multiplication Theorems

Analyzing the Setup

The Logical Bridge

The Final Calculation

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