Probability of Destroying a Bridge with Multiple Bomb Hits: A Case Study

The problem at hand involves calculating the probability of destroying a bridge when multiple bomb hits are made, under the assumption that two hits are required for the bridge to be destroyed. The probability of a single bomb hitting the target is given as 1/5. This article will delve into the step-by-step approach to solving this problem using the principles of probability and the binomial distribution.

Introduction to Binomial Distribution

Binomial distribution is a probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success. In this context, each bomb dropped is a trial, and the success is hitting the target.

Problem Statement

The probability of hitting a target is given as (p frac{1}{5}), and the probability of missing is (q frac{4}{5}). We have six bombs to drop, and the bridge will be destroyed if at least two bombs hit the target. Our task is to calculate the probability of the bridge being destroyed under these conditions.

Step-by-Step Solution

Step 1: Calculate the Probability of Hitting 0 and 1 Bomb

First, we calculate the probability of hitting 0 bombs:

(P(X 0) binom{6}{0} left(frac{1}{5}right)^0 left(frac{4}{5}right)^6 left(frac{4}{5}right)^6 approx 0.262144)

Next, we calculate the probability of hitting 1 bomb:

(P(X 1) binom{6}{1} left(frac{1}{5}right)^1 left(frac{4}{5}right)^5 6 cdot frac{1}{5} cdot left(frac{4}{5}right)^5 6 cdot frac{1}{5} cdot 0.32768 approx 0.393216)

Step 2: Calculate the Probability of Hitting at Least 2 Bombs

To find the probability that the bridge is destroyed, we need to calculate the probability of hitting at least 2 bombs. This can be found by subtracting the probabilities of hitting 0 and 1 bomb from 1:

(P(X geq 2) 1 - P(X 0) - P(X 1) approx 1 - 0.262144 - 0.393216 approx 0.34464)

Therefore, the probability that the bridge is destroyed is approximately (0.34464).

Conclusion

Using the principles of probability and the binomial distribution, we have calculated the probability that the bridge will be destroyed if at least two bombs out of six hit the target. The final answer, given the probability of hitting a target with each bomb is (frac{1}{5}), is approximately (0.34464).

In practical scenarios, such calculations are crucial for military planning and risk assessment, ensuring that strategic targets are effectively neutralized. Understanding and applying these concepts can help in making more informed decisions in similar situations.

Keywords

probability, binomial distribution, bomb targeting, bridge destruction