Animated Solution for Mathematics - Probability: Four persons independently solve a certain problem correctly with probabilities $1/2,3/4,1/4,1/8$. Then the probability that the problem is solved correctly by at least one of them is

Mastering 'At Least One' Problems

Imagine you are sitting in the exam hall, the clock is ticking, and you encounter a probability problem involving four individuals, $A$, $B$, $C$, and $D$, each with their own unique chance of success. The problem asks for the probability that 'at least one' of them solves the problem.

Your first instinct might be to calculate the probability of one person solving it, then two, then three, and finally all four. But stop! Take a deep breath.

In the world of competitive exams like JEE, time is your most precious resource. If you find yourself staring down a path that requires fifteen separate calculations, you are likely missing a hidden, elegant shortcut.

The phrase 'at least one' is a massive red flag in probability—in the best way possible. It is a signal to use the Complement Rule.

The Complement Rule states that the probability of an event occurring is equal to one minus the probability of it not occurring. Mathematically, we write this as:

Think about it: the only way 'at least one' person fails to solve the problem is if 'nobody' solves it. By shifting our focus from the complex 'at least one' scenario to the singular, clean 'nobody' scenario, we transform a mountain of work into a molehill.

Let us break down the failure probabilities. If person $A$ has a success probability $P(A)=\frac{1}{2}$, their failure probability is $P(A^{\prime })=1-\frac{1}{2}=\frac{1}{2}$.

We repeat this for everyone: $P(B^{\prime })=1-\frac{3}{4}=\frac{1}{4}$ $P(C^{\prime })=1-\frac{1}{4}=\frac{3}{4}$ $P(D^{\prime })=1-\frac{1}{8}=\frac{7}{8}$

Now, because the problem states that these individuals are working independently, we can use the multiplication rule for independent events. The probability that absolutely nobody solves the problem is the product of their individual failures:

Substituting our values, we get:

Multiplying the numerators gives us $1\times 1\times 3\times 7=21$, and the denominators give us $2\times 4\times 4\times 8=256$. Thus:

We are almost at the finish line. We return to our master equation: $P(\text{At least one})=1-P(\text{None})$.

Substituting our result, we have:

To subtract these, we express $1$ as $\frac{256}{256}$, leading us to:

There it is! The elegance of the complement rule has saved us from a tedious, error-prone calculation. Remember, in JEE, it is not just about knowing the formulas; it is about recognizing the structure of the problem and choosing the path that leads to the answer with the most efficiency and grace.