Applying the Binomial Distribution in Military Target Shooting

In military operations, precision shooting is a critical skill. This article demonstrates how to apply the binomial distribution to calculate the probability of a soldier hitting a target in a scenario where they fire 8 shots with a 3/4 probability of hitting the target per shot. Let's explore the calculations and their significance.

Theoretical Background

Before diving into the calculations, it's essential to understand the theoretical background. The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials. Each trial is independent, meaning the outcome of one trial does not affect the outcome of another.

Scenario Overview

A soldier performs a series of shooting exercises, firing 8 shots at a target. For each shot, the probability of hitting the target is 3/4. We're interested in three specific probabilities:

The probability that the soldier hits the target exactly once. The probability that the soldier hits the target exactly 4 times. The probability that the soldier hits the target exactly 5 times.

Calculation Process

The probability of having exactly k successes in n trials can be calculated using the binomial probability formula:

P(X k) binom{n}{k} p^k (1-p)^{n-k}

Where:

n is the total number of trials (shots fired). k is the number of successful trials (hits). p is the probability of success on each trial (hitting the target). binom{n}{k} is the binomial coefficient, the number of ways to choose k successes in n trials.

Given Parameters

In this scenario:

n 8 (the soldier fires 8 shots). p frac{3}{4} (probability of hitting the target). 1 - p frac{1}{4} (probability of missing the target).

Probability of Hitting the Target Exactly Once (k 1)

P(X 1) binom{8}{1} left(frac{3}{4}right)^1 left(frac{1}{4}right)^7

8 cdot frac{3}{4} cdot frac{1}{16384} 8 cdot frac{3}{65536} frac{24}{65536} frac{3}{8192}

Probability of Hitting the Target Exactly 4 Times (k 4)

P(X 4) binom{8}{4} left(frac{3}{4}right)^4 left(frac{1}{4}right)^4

70 cdot left(frac{3}{4}right)^4 cdot left(frac{1}{4}right)^4

70 cdot frac{81}{256} cdot frac{1}{256} 70 cdot frac{81}{65536} frac{5670}{65536}

Probability of Hitting the Target Exactly 5 Times (k 5)

P(X 5) binom{8}{5} left(frac{3}{4}right)^5 left(frac{1}{4}right)^3

56 cdot left(frac{3}{4}right)^5 cdot left(frac{1}{4}right)^3

56 cdot frac{243}{1024} cdot frac{1}{64} 56 cdot frac{243}{65536} frac{13608}{65536}

Summary of Probabilities

P(X 1) approx 0.000366

P(X 4) approx 0.0862

P(X 5) approx 0.2079

Conclusion

These detailed calculations highlight the importance of understanding the binomial distribution for simulating and evaluating shooting accuracy in military and similar contexts. These probabilities provide a framework for assessing the likelihood of various outcomes in shooting scenarios.