Animated Solution for Mathematics - Probability: Four persons can hit a target correctly with probabilities $1/2,1/3,1/4$ and $1/8$ respectively. if all hit at the target independently, then the probability that the target would be hit, is

Imagine you are standing at a shooting range. There are four people—let us call them $A$, $B$, $C$, and $D$—all aiming at the exact same target. They are all going to take a shot independently.

This is the crucial part: one person's performance has no bearing on the others. We have four distinct individuals, each with their own unique skill level, trying to hit a mark. Our goal is to find the probability that the target is hit at least once.

We are given their individual hitting probabilities: $P(A)=\frac{1}{2}$, $P(B)=\frac{1}{3}$, $P(C)=\frac{1}{4}$, and $P(D)=\frac{1}{8}$.

Person $A$ is clearly the marksman of the group, while $D$ is still learning the ropes. If we tried to calculate the probability of the target being hit by summing up every possible way it could happen, we would be here all day. It is a combinatorial nightmare.

Whenever you encounter the phrase "at least one" in a probability problem, a lightbulb should go off in your head. It is almost always easier to calculate the probability of the opposite event and subtract it from one.

The opposite of "the target is hit at least once" is "the target is not hit at all." This is the complement rule:

This elegant strategy is a favorite in JEE exams because it rewards conceptual clarity over brute-force calculation.

To find the probability that nobody hits the target, we first need to know the probability that each individual misses. For any person, the probability of missing is simply $1$ minus their probability of hitting.

Let us calculate these one by one:

For person $A$: $P(A^{\prime })=1-P(A)=1-\frac{1}{2}=\frac{1}{2}$.

For person $B$: $P(B^{\prime })=1-P(B)=1-\frac{1}{3}=\frac{2}{3}$.

For person $C$: $P(C^{\prime })=1-P(C)=1-\frac{1}{4}=\frac{3}{4}$.

For person $D$: $P(D^{\prime })=1-P(D)=1-\frac{1}{8}=\frac{7}{8}$.

Notice how the miss probabilities increase as the skill levels decrease. This makes perfect physical sense.

Because the shots are independent, the probability that all four people miss simultaneously is the product of their individual miss probabilities. We do not need to worry about complex interactions.

We simply multiply:

Substituting our values, we get:

Now, watch the magic of cancellation. The $2$ in the numerator cancels with the $2$ in the denominator, and the $3$ in the numerator cancels with the $3$ in the denominator. We are left with:

We have found the probability that everyone misses is $\frac{7}{32}$. To find the probability that the target is hit at least once, we subtract this from $1$:

And there it is! The probability that the target is hit is $\frac{25}{32}$. It is a high probability, which makes sense given that four people are firing. By using the complement rule, we turned a potentially overwhelming problem into a series of simple, elegant steps.