Probability of Hits and Misses in a Target-Shooting Test

In a target-shooting test, the probability of hitting the target is 1/2 for shooter A, 2/3 for shooter B, and 3/4 for shooter C. This article aims to calculate the probability of specific outcomes when all three shooters fire at the target. Let's explore the probability that none of them hit the target and the probability that at most two of them hit the target.

None of Them Hit the Target

The probability of none of them hitting the target is determined by calculating the product of the probabilities of each person missing the target. The probability that shooter A misses the target is 1 - 1/2 1/2, for shooter B it is 1 - 2/3 1/3, and for shooter C it is 1 - 3/4 1/4. Therefore, the overall probability that none of them hit the target is:

1 - 1/2?1/3?1/41/2 * 1/3 * 1/41232?13777?1334

1/2 * 1/3 * 1/4 1/24

At Most Two of Them Hit the Target

The probability that at most two of the three shooters hit the target is the complement of the event that all three hit the target. We start by calculating the probability that all three hit the target. This is given by the product of their individual probabilities of hitting:

P(all hit) 1/2 * 2/3 * 3/4 1/4

Therefore, the probability that at most two of them hit the target is:

1 - P(all hit) 1 - 1/4 3/4

Discussion

The results show that the probability of none of them hitting the target is 1/24, and the probability that at most two of them hit the target is 3/4. This is because the complementary event (at least three shooters hitting) occurs with a probability of 1/4, so the rest (75%) must be the probability that at most two hit the target. Even though each shooter has a different skill level, the overall outcome heavily depends on the collective probability.

It is clear that the skill level and the size of the target play a significant role in the overall outcome of the shooting. While some individuals may hit the target more often than others, the probability calculation provides a mathematical framework to analyze and predict the outcomes of such scenarios.