Determine if a specific number of people can be infected after a series of random group interactions during a zombie outbreak | Step-by-Step Solution

TinyProf's Guide to the Zombie Plague Problem! 🧟‍♂️

What We're Solving:

We need to determine whether it's possible for exactly 100 people to be infected after 10 nights in a town of 3000, where people randomly form groups of 3 each night and zombies spread within these groups.

The Approach:

This is a fascinating probability problem that combines infection modeling with group theory! We'll analyze this by understanding the infection dynamics and mathematical constraints. The key insight is recognizing what makes this problem tractable versus intractable.

Step-by-Step Solution:

Step 1: Understand the Infection Mechanism

• Each night, 3000 people form 1000 random groups of 3
• Within each group, if ≥1 person is a zombie, ALL non-zombies become zombies
• This means infection can only increase or stay the same each night (never decrease)

Step 2: Analyze the Mathematical Structure The crucial insight is that this problem has enormous complexity:

• The number of ways to partition 3000 people into groups of 3 is astronomical
• The infection spread depends on these random groupings
• Each night's outcome affects all subsequent nights

Step 3: Consider Boundary Cases

• If we start with Z zombies, we'll end with ≥Z zombies after any number of nights
• The infection spreads based on group overlap probabilities
• With 100 initial infections and random mixing, the spread will be probabilistic

Step 4: Recognize the Problem Type This is actually asking about the reachability of specific states in a complex stochastic process. The question "Can exactly 100 people be infected after 10 nights?" requires us to consider:

• Starting configurations that could lead to this outcome
• The probability dynamics of group formation
• Whether there exist paths through the probability space to reach exactly 100 infected

Step 5: Apply Probabilistic Reasoning For exactly 100 to be infected after 10 nights:

• We'd need a starting configuration with ≤100 infected
• The infection would need to either stay constant or grow then somehow stabilize
• Given random group formation, complete containment becomes increasingly unlikely

The Answer:

This problem is computationally intractable to solve exactly, but we can reason that having exactly 100 infected after 10 nights is theoretically possible but extremely unlikely.

Here's why: If we start with fewer than 100 infected people spread strategically, there exist (albeit rare) sequences of random groupings that could result in exactly 100 infected after 10 nights. However, the probability of such specific outcomes approaches zero as the number of nights increases due to the random mixing.

The more practical answer is that while mathematically possible, it's so improbable that for realistic purposes, we'd expect either much faster spread or different final numbers.

Memory Tip:

Think of this like trying to get exactly the right amount of food coloring in a glass of water after 10 rounds of stirring and adding drops - technically possible, but the randomness makes precise outcomes nearly impossible! 🎨

Great job tackling such a complex probability problem! These types of questions help us understand how randomness and constraints interact in real-world scenarios.