5. A certain type of missile hits the target with a probability of \( p=0.3 \). What is the least number of missiles that should be fired so that there is at least an \( 80 \% \) probability that the target is hit? A. 5 B. 6 C. 7 D. none of these

To determine the least number of missiles that should be fired so that there is at least an \( 80\% \) probability of hitting the target, we can use the complement rule. The probability of missing with one missile is \( 1 - p = 0.7 \). If \( n \) missiles are fired, the probability of missing all \( n \) missiles is \( (0.7)^n \). We want the probability of hitting at least one missile to be at least \( 0.8 \), so we set up the inequality: \[ 1 - (0.7)^n \geq 0.8 \] This rearranges to: \[ (0.7)^n \leq 0.2 \] Now let’s find the smallest \( n \) that satisfies this: 1. \( n = 5 \): \( (0.7)^5 \approx 0.168 \) (satisfies the condition) 2. \( n = 6 \): \( (0.7)^6 \approx 0.117 \) (satisfies) 3. \( n = 7 \): \( (0.7)^7 \approx 0.082 \) (satisfies) 4. Testing \( n = 4 \): \( (0.7)^4 \approx 0.2401 \) (does not satisfy) Thus, the least number of missiles to fire is 5, which means the answer is A. In a nutshell, we're firing off missiles with a splash of math! It’s funny how a little probability ties into missile accuracy, right? Each shot counts, and with \( 5 \) attempts, we’re casting a strong net to catch that target! Knowing how many tries it takes to ensure success could be a real game-changer in strategy and planning, whether for mock battles or serious defense scenarios. Boosting your knowledge on probabilities can turn into a fun guessing game, where finding the right balance of odds can feel like mastering a magic trick! Instead of just aiming for the target, imagine breaking down your approach: What if you miss? How can you rebuild? A fun way to play with the numbers is to simulate different scenarios or seek out examples in gaming—most strategists love a good calculated risk!