Understanding the Probability of Hitting a Target Exactly Five Times in 10 Shots

When engaging in activities such as target shooting, the question of the probability of achieving a particular outcome can be both intriguing and informative. For instance, if a person fires 10 shots at a target and the probability of hitting the target is 3/5, one might wonder: What is the probability that the target will be hit exactly five times? This article will delve into the mathematical analysis to provide a comprehensive answer.

Introduction to the Problem

The problem can be broken down into understanding the probability of hitting a target under a specific number of trials, where the success (hitting the target) probability is known. Given that the probability of hitting the target (PH) is 3/5 or 0.6, and the probability of missing (PM) is 1 - 0.6 or 0.4, we can apply the principles of the binomial distribution to find the answer.

The Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials. In this scenario, each shot is an independent trial with two possible outcomes: success (hitting the target) or failure (missing the target). The parameters involved are:

n 10, the total number of trials (shots). p 0.6, the probability of success (hitting the target). q 1 - 0.6 0.4, the probability of failure (missing the target).

Calculating the Probability

Using the binomial distribution formula, we can calculate the exact probability of hitting the target exactly five times. The probability mass function for a binomial distribution is given by:

P(X) x b(n, x)p^xq^{n-x}

Where:

x is the number of successes (in this case, 5). b(n, x) is the binomial coefficient, which can be calculated using the formula 10C5 or (n! / (x!(n-x)!)). p is the probability of success (0.6 in this case). q is the probability of failure (0.4 in this case).

Step-by-Step Calculation

Calculate the binomial coefficient 10C5: 10C5 10! / (5!(10-5)!) 10! / (5!5!) 252 Calculate the probability of success (hitting the target) 5 times: 0.6^5 0.07776 Calculate the probability of failure (missing the target) 5 times: 0.4^5 0.01024 Combine all the values: P(X5) 252 * 0.07776 * 0.01024 ≈ 0.200658

Conclusion and Validation

The probability that the target is hit exactly five times in 10 shots is approximately 0.200658. This calculation can be validated through software or hand calculations, ensuring the accuracy of the result.

Final Thoughts

Exploring probability in such scenarios not only enriches our understanding of mathematics but also aids in making informed decisions in various real-world applications. Whether you are a student studying probability, a professional in risk management, or someone interested in statistics, this problem provides a solid foundation.