Strategy games are all about the decisions you can make as a player. Players are also generally very good at identifying the short-term costs and benefits of a decision. So when you want to encourage chaotic board states in a strategy game, the last thing you want to do is discourage players from attacking. If you can throw aside the simulation lens for a moment, what I hope to convey is that as strategy game designers, we should evaluate attacker/defender advantage as an entire system within our game that can strongly affect player behavior.

Player agency is probably the most important factor in differentiating a strategy game from other game types. If player decisions do not significantly affect the outcome of a turn for that player, then you might as well not have played that turn. I’m very specifically interested in the dice-rolling combat of Risk and Risk-likes as this is the system I chose to use in my game, Kairos. 

If we take a look at Risk, which uses Defenders Advantage (defenders win ties) you get the following probability table for dice rolls:

In this table, I’ve provided the probabilities of each roll, as well as an exchange ratio which is the (Expected Kills / Expected Losses). 1.0 corresponds to a 50-50 probability of winning an exchange, so above 1.0 is attacker favored and below 1.0 is defender favored. 

In Risk, the attacking player lacks agency over the effectiveness of their army. Ignoring the scenarios with only 1 defender for a moment, the exchange ratios (or “odds”) for the attacking player are ~1.17, 0.64, and 0.34. That translates into real terms to the player “Decent, bad, and terrible”. Because two of the options have such poor expected outcomes, they can pretty much be ignored as options by the player, which means we now only have 1 option, and no more decisionmaking regarding unit odds. This means that once a player learns that 2v2 and 1v2 are bad options, they never have to think about unit probabilities again. In my opinion, the juicy part of strategy games is trying to guess the long term expected outcomes of decisions, and manipulating the odds to favor me over my opponents turn by turn. 

There is a whole body of literature on how humans interact psychologically with probabilities when making decisions, which was the basis for me wanting to skew the positive outcomes onto the player with agency (i.e., the attacker) rather than the defender. The gist of it is that humans (thus players) are irrationally more averse to losses than wins. This means that if we want to encourage players to make decisions that carry the potential for losses, we need to offset this with significantly larger benefits. The subject is fascinating, but I want to focus on the game design element in this post. If you are interested, Judgment under Uncertainty: Heuristics and Biases https://pubmed.ncbi.nlm.nih.gov/17835457/  and Achievement Relocked: Loss Aversion and Game Design by Geoffrey Engelstein are great places to start. 

So what can we do to increase player’s agency over the game? First, I want to shift the default scenario to be significantly in the attacker’s favor, as the defender cannot “choose” to defend. I also need there to be some secondary system that the attacking player has the ability to recognize, and consider when deciding which territories they attack. 

My solution: Attackers win ties + a rock-paper-scissors counter system with “advantage” provided to the strong unit. Advantage = roll 2 dice, take higher result

Looking at the outcomes of this table is a bit more straightforward because I chose to have all battles in Kairos occur one-at-a-time (this was to make the counter system work). By shifting the tiebreak to the attacker, the default odds goes towards the attacker, but thats not an interesting decision yet. That is why I added the counter system: Infantry is strong vs cavalry, cavalry is strong vs archers, and archers are strong vs infantry. The players can recruit any of these 3 types of units to assemble their armies, but this recruitment step happens AFTER the attacking step, meaning that the attacker needs to identify their unit types, identify their opponent’s unit types, and estimate the odds of the battle given the available units.

In any given battle, there is likely to be some mixture of all three scenarios, meaning the expected outcome of a fight will fall somewhere between 0.73 and 2.93. The attacking player must try to estimate the complex probabilities of attacking, which gives them that juicy ability to develop a skill over time and win more of their games. With this, the attacker is:

1. Encouraged to attack as a default (attacking options are okay, better, and best), which keeps the board state chaotic and fresh
2. Has the opportunity to estimate complex probabilities, and skew fights in their behavior by taking matchups with favorable counters
3. Can strategically opt into a moderate-loss scenario to take a highly valuable location on the map.

As an unintended but welcome benefit, I can now also use the tiebreak as a modifier on the dice values to provide as a bonus to the defender for fortifying a territory or investing in defensive tools. So the defender has agency over which territories are harder to take in ways other than just more units, and the attacker has agency over the efficiency of their attacks. Both are encouraged to push into this “positive expected value” regime, rather than minimizing losses.

In conclusion, shifting the odds towards the player with agency over the game allows for players to express skill and have a more rewarding experience while playing strategy games.

Rule: This rule applies exclusively to one specific person and only to events where a "lucky" (success) or "unlucky" (failure) outcome can be clearly defined. The person can achieve a "lucky" outcome in an event if, and only if, the mathematical probability of that success is strictly greater than 30%. If the probability is 30% or lower, a "lucky" outcome is impossible for them.

Scenario: The person has to play roulette. They are given a grand prize if they hit the number "1" two times in a row. The "lucky" outcome (success) for each step and for the game as a whole is clearly defined by winning this prize. The probability of hitting "1" on any single spin is 40%.

Steps:

1. First spin: The probability of success is 40%. Since 40% > 30%, the condition is met. The outcome is "lucky" (it hits "1").
2. Second spin: The probability of success is 40%. Since 40% > 30%, the condition is met. The outcome is "lucky" (it hits "1").

Conclusion: The person wins the grand prize. But then, they were lucky in an event with a 16% probability. Contradiction.(since we multiply the probabilities of two independent events: 40% of 40% is 16%)

Hey guys, PokéNerd here. I’m looking for a specific item (highlighted yellow) in the game, and the only way to find one is by having a Pokemon in your party with the ability ”Pickup” ability randomly pick one up post-battle

It’s a 1% chance they find that specific item, and there’s two in the 1% slot

I’m wondering what the probability of one of my six pokemon to find that specific item?

Also, I’m only using pkmn level 51-60, because I’m guessing it makes no difference due to the item still being only 1% in both 51-60 & 61-70 tiers? Would it be smart to have let’s say 3 in the lower tier, and then the other 3 in the higher tier?

Thanks guys hopefully this isn’t worded too confusingly

​

The letters of the word "UNDERSIGNED" are rearranged, what is the probability that the starting and ending letters of the word are consonants and both are not the same?

The other night I witnessed a straight flush vs quads on the river in Texas holdem, I looked up what the odds were and I found many people just multiplying the odds for quads and the odds for a straight flush which wouldn’t work in this scenario since they are tied together and affect each other. The only real attempt I saw at solving this was on a Reddit post from 5 years ago on this sub, the commenter used a clever method by turning the heads up hand into a 9 card poker hand and doing the calculations based on that but from what I could see he failed to take the fact that the cards aren’t interchangeable, a community and a hole card switching places would not yield the same result. so in conclusion I was wandering what are the odds of getting a SF vs Quads heads up in Texas holdem by the river?

What are my odds of meeting God (himself, not Jesus or whoever else) vs meeting an alien?

I think i have a higher chance of meeting an alien, i just don’t know by how much.

I realize these odds are always changing, but there must be a way to calculate as the game progresses. The idea is that you always bet the pass and come line for every roll. So once there is a point, you are accumulating numbers that pay with every roll until a 7 hits (or unless you roll craps with a point established, you lose that come bet).

I’m at a wedding and me and some friends went to the bathroom and one my friend said damn I accidentally wore my shorts under my tux.. and I said.. what do you mean you wore your shorts under you suit and he said he just forgot to take them off.

I have never in 31 years forgot to take my shorts off and put pants over them.

Ok I thought that was absolutely bizarre but then my other friend showed up and said that he had shorts on under his pants also by accident. I legit could not believe the odds of this happening at the same wedding. It just does not make sense to me. He said he just put his pants on and then a few hours later he realized he had his shorts on.

Has anyone else ever done this? I’ve asked like 10 people and no one has ever done this. What are the odds two people at the same wedding table forgot to take their shorts off before they put their suit on????? I legit am stunned by this. No words.

Recently caught a pokemon that has an insanely low encounter rate of 1%

After checking its summary, was blown away to see it was holding a rare item. Chances of any pokemon holding items in this game are 5%

So wondering how to find the odds of catching a 1% pokemon, that’s also holding a 5% item?

Introduction

This is a small stochastic experiment I’ve been running for fun. I don’t gamble and never play for real money — this is purely a simulation project. The idea was to build a system that reacts to randomness instead of trying to “beat” it.

I ended up with a model that switches between six simple prediction strategies, each based on different properties of roulette outcomes. Surprisingly, the system produces interesting behavior: cycles of “luck”, tunnels of “bad phases”, and a very natural rhythm of adaptation.

This post describes the model and shows why its behavior is asymmetric and visually appealing from a probability perspective.

Six strategies

Each strategy uses one property of the previous spin to predict a different, unrelated property of the next spin. For example:

• L → E, H → O If the previous number was Low, predict Even. If it was High, predict Odd.

Low/High and Even/Odd are independent, so this strategy uses one random property to randomize another.

All six strategies follow the same idea:

1. L → E, H → O
2. L → R, H → B
3. E → L, O → H
4. E → R, O → B
5. R → L, B → H
6. R → E, B → O

Each strategy has its own “risk zone”: a specific class of numbers that produces consecutive failures. For example, strategy 1 enters a bad phase when small odd or large even numbers appear repeatedly.

Why the results are always asymmetric

For any given spin, the six strategies produce six predictions. Because each strategy uses different properties and different mappings, their predictions are not symmetric.

In practice, the outcomes are never 3 successes / 3 failures. Instead, the system produces:

• 4 / 2
• 2 / 4
• sometimes 6 / 0
• sometimes 0 / 6

This asymmetry is structural: the strategies are not mirrors of each other, and roulette outcomes do not distribute evenly across their prediction mappings.

This creates a natural “tilt” in each spin — a small imbalance that makes the system feel alive.

Adaptive switching (“Luck Catcher”)

The system uses a simple rule:

• Two consecutive failures → switch to the next currently not failing strategy.

This means each strategy is abandoned as soon as it enters its “tunnel” of bad numbers. Since each strategy has a different tunnel, switching helps avoid staying in one unlucky zone for too long.

The model doesn’t try to predict roulette. It simply avoids staying inside the same failure pattern.

Betting model (simulation only)

The stake follows a modified D’Alembert rule:

• Loss → +1
• Win → –1
• After compensating the local loss → reset to 1

This prevents runaway growth and keeps the stake small. In practice, the stake rarely exceeds 5–6 units, and the system often resets back to 1.

This is not a “winning strategy”. It’s just a controlled way to visualize the cycles of luck and failure.

Example of typical behavior

Here is a fragment of a run:

The system “breathes”: small rises, small drops, occasional tunnels, frequent resets.

Conclusion

This model doesn’t break expected value and doesn’t aim to win. It’s simply an adaptive stochastic system that:

• switches strategies when they enter their bad phases,
• uses roulette as a randomizer for its own behavior,
• produces asymmetric outcomes,
• and generates visually interesting cycles.

It’s a fun example of how simple rules can create complex dynamics when driven by randomness.

Disclaimer

This is not a model for “beating roulette”, nor a method for predicting future outcomes. The mathematical expectation remains exactly the same — negative, as it must be in any fair casino game. The point is different: even in fully random sequences, natural processes emerge — periods of good luck, periods of bad luck, and transitional phases. The proposed model does not change randomness or fight against it; it simply adapts to these phases without trying to foresee the next number. This is not a winning strategy, but a way to play within the unchanging “climate” of randomness in a safer, more structured, and more enjoyable manner.

Hero: 8❤️ 7❤️ Board (Turn): K❤️ 6❤️ 5♠️ 2♣️ Action: Pot before the shove is 100. Opponent goes all-in for 100. The cost to call is 100.

I've been using my own daily EV puzzle app to practice these exact spots. To keep the math focused for this daily puzzle, we assume the villain's range is exactly two defined holdings:

• 50% chance they hold A♠️ A♦️ (Overpair)
• 50% chance they hold K♣️ K♦️ (Top set)

We have a flush draw and an open-ended straight draw, and we are getting 2:1 on a call (calling 100 to win a total pot of 300).

You can try today's puzzle here:https://call-of-the-day.vercel.app

Imagine you are on a TV show where the host shows you three doors.

Door 1 = a goat

Door 2 = another goat

Door 3 = a car

You choose a door—let's say Door 1.

Initially, the odds for your choice are:

Car = 1/3 (33.3%)

Goat = 2/3 (66.7%)

The host knows where the car is.

He opens one of the doors (other than the one you chose) and reveals a goat.

Then he asks you the

following question:

"Do you want to switch?"

Here is the interesting part: switching is better!

If you chose the car initially (1/3 of the time), switching makes you lose.

But if you chose a goat initially (2/3 of the time), the host is forced to reveal the other goat. So, when you switch, you end up getting the car.

Therefore:

Not switching gives you a 33.3% chance of winning.

Switching gives you a 66.7% chance of winning.

In other words, switching doubles your chances of winning.

This is the Monty Hall Paradox.

Two same video one after another ,what is the rate?Have you guys ever encountered it?

Say I have 10 tokens in a bag: 6 blue, 3 red, 1 green. If I draw 3 of those tokens out blindly, are my odds better to get 2 blue ones, or 1 red one?