Animated Solution for Mathematics - Probability: $A$ and $B$ are events such that $P(A\cup B)=3/4,P(A\cap B)=1/4,P(\bar{A})=2/3$ then $P(\bar{A}\cap B)$ is

Welcome, students. Today, we are going to explore a beautiful problem in probability. Imagine a sample space, which we will represent as a bounding rectangle.

Inside this rectangle, we have two events, $A$ and $B$. We are given the following information: $P(A\cup B)=\frac{3}{4}$ $P(A\cap B)=\frac{1}{4}$ * $P(\bar{A})=\frac{2}{3}$

Let's set up our visual stage and solve this step by step.

Our first goal is to find the probability of event $A$ happening. We are given the probability of $A$ bar, denoted as $P(\bar{A})$.

Since an event either happens or it does not, the sum of these probabilities is always exactly one. We use the complement rule:

Substituting our given value:

This is our first solid building block.

The Bridge

Moving forward, we need to find the probability of event $B$. To do this, we use the Addition Theorem of probability:

We subtract the intersection because adding $A$ and $B$ counts the overlapping region twice. We are given $P(A\cap B)=\frac{1}{4}$, which represents this crucial overlap.

Now, substitute all known values into the Addition Theorem:

To isolate $P(B)$, we rearrange the terms:

Since $\frac{3}{4}+\frac{1}{4}=1$, the expression simplifies to:

Now, we calculate the target probability: $P(\bar{A}\cap B)$. Physically, this represents the region where event $B$ occurs, but event $A$ does not.

Visually, this is the region strictly inside $B$ excluding the intersection with $A$. The formula is:

Substituting our calculated values:

Using the common denominator of $12$:

The final answer is $\frac{5}{12}$. Probability is not just about numbers; it is about understanding the geometry of chance.