Animated Solution for Mathematics - Probability: $E$ and $F$ are two independent events. The probability that both $E$ and $F$ happen is $1/12$ and the probability that neither $E$ nor $F$ happens is $1/2$. Then,

Unlocking Probability

Welcome, fellow traveler on the JEE journey. Today, we are not just solving a probability problem; we are uncovering the hidden symmetry of independent events.

Imagine you are standing before a Venn diagram with two circles, $E$ and $F$. They are independent, which is a fancy way of saying that the occurrence of one tells you absolutely nothing about the occurrence of the other. It is a state of perfect, blissful ignorance between two events.

The Variables of Uncertainty

Let us define our players. We assign $P(E)=x$ and $P(F)=y$.

The problem provides the probability that both happen:

Because $E$ and $F$ are independent, we know that the probability of their intersection is simply the product of their individual probabilities. Thus, we have our first anchor equation:

The Secret of the Complements

The problem also states that the probability that neither $E$ nor $F$ happens is $\frac{1}{2}$. In the language of sets, this is $P(E^c\cap F^c)=\frac{1}{2}$.

If $E$ and $F$ are independent, then their complements, $E^c$ and $F^c$, are also independent. Therefore, we can express the intersection of the complements as the product of their individual probabilities:

Since $P(E^c)=1-x$ and $P(F^c)=1-y$, we obtain the following relationship:

The Algebraic Bridge

We now have a system of two equations: $xy=\frac{1}{12}$ and $(1-x)(1-y)=\frac{1}{2}$. Let us expand the second equation:

This simplifies to the form:

Substituting our known value $xy=\frac{1}{12}$ into this expression, we get:

Solving for the sum $(x+y)$, we find:

The Quadratic Finale

We are left with a classic scenario: we know the sum of two numbers is $\frac{7}{12}$ and their product is $\frac{1}{12}$. These values are the roots of the quadratic equation $t^2-St+P=0$.

Substituting our values, we get:

Multiplying by $12$ to clear the fractions, we obtain:

Factoring the quadratic equation:

The roots are $t=\frac{1}{3}$ and $t=\frac{1}{4}$. Thus, the probabilities are $P(E)=\frac{1}{3}$ and $P(F)=\frac{1}{4}$ (or vice versa). You have successfully navigated the logic of independence and emerged victorious.