Probability in Game Mechanics

Discrete vs. Continuous: Choosing the Right Model

First, it's important to understand the distinction between discrete and continuous random variables.

• Discrete variables are countable. Think "number of...": number of loot drops, number of successful hits, number of units produced. The outcome is always an integer.
• Continuous variables can take any value within a given range. Think measurements: time to kill, damage dealt, a unit's speed.

Part 1: Modeling "How Many?" with the Binomial Distribution

Let's start with a very common scenario in strategy games: a unit attacks multiple times, and each attack has a certain probability of success. How do we model the total number of successful hits?

This is the job of the Binomial distribution. It applies whenever you have a situation that meets four criteria:

1. A fixed number of trials (e.g., 10 arrows are fired). Let's call this $n$.
2. Each trial is independent (one arrow hitting doesn't affect the next).
3. Each trial has only two possible outcomes (hit or miss).
4. The probability of success is the same for each trial (the archer has a 70% chance to hit with each arrow). Let's call this $p$.

The binomial distribution answers the question: "What is the probability of getting exactly $k$ successes in $n$ trials?"

To build a strong intuition for this, let's watch a video that breaks down the core concepts from the ground up.

As the video explained, the formula for the binomial distribution can look intimidating, but it's just telling that two-part story:

• $p^k(1-p{)}^{n-k}$ is the probability of one specific sequence of $k$ successes and $n-k$ failures. Given your background, you can think of this as the probability of a specific build order playing out in one exact way.
• $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ is the number of ways to get $k$ successes in $n$ trials. This is called the "binomial coefficient" or "n choose k". It accounts for the fact that we don't care about the order of the hits and misses, just the total.

To really cement the "number of ways" part, the next video provides a fantastic visual explanation.

Application to Game Balance: Drop Rates and Critical Hits

Now, let's bring this directly into game balancing. You're designing a system where a player opens 10 chests, and each chest has a 20% chance of containing a rare gem. What is the probability of getting exactly 3 gems?

• $n=10$ (number of chests)
• $p=0.2$ (probability of a gem)
• $k=3$ (desired number of gems)

Plugging this into the formula gives us a probability of about 20.1%. Knowing the probability of every outcome allows you to design and tune the player's reward experience with precision.

The following reading shows how this is used to think about loot drop systems.

The most powerful aspect for a designer is seeing how the distribution's shape changes as you tweak the probability, p.

Part 2: Modeling Continuous Outcomes with the Normal Distribution

What about damage rolls, ability cooldowns, or AI reaction times? These are continuous variables. While you could use a simple uniform distribution (like a standard dice roll where every outcome is equally likely), many natural and game phenomena tend to cluster around an average. The perfect tool for this is the Normal distribution, also known as the "bell curve".

The Normal distribution is defined by just two parameters you are already familiar with:

• The mean ($\mu$), which sets the center of the curve.
• The standard deviation ($\sigma$), which sets the spread or width of the curve.

This direct link to our previous lesson makes it an incredibly powerful concept.

Let's return to the video series to get an overview.

Application to Game Balance: Damage Rolls

The 68-95-99.7 rule is a fantastic rule of thumb for a game designer. If you're designing a rifle with damage that is normally distributed with a mean of $\mu =100$ and a standard deviation of $\sigma =15$, you immediately know:

• About 68% of shots will deal damage between 85 and 115 ($\mu \pm 1\sigma$).
• About 95% of shots will deal damage between 70 and 130 ($\mu \pm 2\sigma$).
• Almost all shots (99.7%) will be between 55 and 145 ($\mu \pm 3\sigma$).

This gives you a robust mental model for the weapon's consistency. A smaller $\sigma$ means a more reliable weapon; a larger $\sigma$ means a more random, "swingy" one.

For more precise questions ("what's the chance of doing more than 130 damage?"), you use the Z-score, as shown in the video. The Z-score is a universal translator that rephrases the question in terms of standard deviations from the mean:

For our rifle, the Z-score for 130 damage is $(130-100)/15=2$. This means 130 damage is exactly 2 standard deviations above the mean. By looking this up in a Z-table (or using a calculator), you can find the exact probability. In this case, it's about 2.28%.

The paper below gives some simple game-related examples for these distributions.

A Glimpse at Other Useful Distributions

While Binomial and Normal are the two most fundamental distributions, your work in RTS and Grand Strategy games will often involve other patterns of randomness. The video you watched mentions a few others that are highly relevant:

• Exponential Distribution: Models the time between events, like how long until the next resource harvester arrives or an enemy squad spawns. If the average rate of spawning is known, this distribution tells you the probability of the wait time.
• Poisson Distribution: Models the number of rare events happening over a fixed interval of time or space. For example, how many meteorites strike a map sector in a 5-minute period, given an average rate.
• Negative Binomial Distribution: This is a crucial one for player experience. Instead of asking "how many drops in 10 chests?", it asks "how many chests must I open to get 3 drops?". This is the math behind "pity timers" in games, which prevent extreme bad luck by putting an upper bound on how long a player has to wait for a reward. The Red Blob Games article you read (LINK) introduces this in its later sections if you are curious.

Don't worry about mastering these right now; the goal is to be aware that there is a whole family of distributions, each telling a different story about randomness.

Summary and Next Steps

Today you've gained a powerful new way to model and control randomness in your games. You've moved beyond simple averages to describing the entire range of possibilities for a given mechanic.

Key Takeaways:

• Probability distributions describe the likelihood of every possible outcome of a random event.
• The Binomial distribution is your tool for modeling discrete "how many successes in n trials?" scenarios, like critical hits or successful dodges. It's defined by the number of trials ($n$) and the probability of success ($p$).
• The Normal distribution (bell curve) is your tool for modeling continuous outcomes that cluster around an average, like damage rolls. It's defined by the mean ($\mu$) and standard deviation ($\sigma$).
• The shape of a distribution is a design lever. By changing parameters like $p$ or $\sigma$, you directly control the consistency and feel of your mechanics.

In our next lesson, we will build directly on this foundation to explore expected value. The expected value is the mean of a probability distribution, representing the average outcome you'd expect if you ran the random event countless times. This will give us a single, powerful number to compare the long-term value of different strategic choices, like whether to build an economic or a military unit.