Probability Calculation for At Least One Target Being Hit by Multiple Shooters

In probability theory, the concept of independent events is pivotal when determining the likelihood of an event occurring. Consider three shooters, A, B, and C, each with varying probabilities of hitting a target. If A, B, and C have probabilities of 0.3, 0.4, and 0.5, respectively, to hit the target, the question arises: what is the probability that at least one of them hits the target?

Understanding the Underlying Probability Theory

The problem can be approached by first calculating the probability that none of the shooters hits the target. Each shooter has a unique probability of missing the target, which can be expressed as the complement of their hitting probability. Let's denote the probability of A, B, and C missing the target as PA’, PB’, and PC’, respectively:

The probability that all three miss the target is the product of their individual probabilities of missing:

Pr(nobody hits the target) PA’ PB’ PC’ 0.7 0.6 0.5 0.21

Solving for the Desired Probability

To find the probability that at least one of the shooters hits the target, we need to calculate the complement of the probability that none of them hits it:

Pr(at least one hits the target) 1 - Pr(nobody hits the target) 1 - 0.21 0.79 or 79%

Verification Through All Combinations

To reiterate, we can verify this by adding up the probabilities of all the different outcomes where at least one shooter hits the target. Despite there being seven different ways (A, B, C, AB, AC, BC, ABC), the simpler approach is to calculate the probability that none of them hit and subtract it from 1:

1 - (1 - 0.3)(1 - 0.4)(1 - 0.5) 1 - 0.21 0.79 79%

This confirmation ensures the accuracy of the initial result, reaffirming that the probability that at least one of the three shooters hits the target is 79%, given their respective probabilities of hitting the target are 0.3, 0.4, and 0.5.

Implications and Applications

Understanding this concept is crucial in various applied fields, including reliability engineering, risk management, and data analysis. The calculation of such probabilities helps in assessing the effectiveness of different systems and strategies in achieving a target outcome. For example, in defense systems, the probability of hitting a target is critical for mission success.