r/probabilitytheory

Most players think I’m crazy when I offer them $750 for a single Light Blue or Brown, or when I trade them a high-value Dark Blue or Green single just for a Brown or Light Blue card that doesn't even complete a monopoly for me.

They think they’re fleecing me because the trade doesn't even complete a set for me.

But they don't see the math. I am not buying a property; I am buying probability. By securing just one piece of these cheap sets early, I mathematically rig the rest of the game in my favor so I never have to make a risky set-for-set trade. Here is the exact logic and the numbers to back it up.

1. The Math: Upgrading Your Odds

If you rely on the dice to hand you a full monopoly, you are going to lose. Look at the attached probability data for a standard 4-player game over 20 rounds:

The Brown Problem: The odds of you landing on both Browns (2 of 2) naturally are a dismal 4.5%.

The Light Blue Problem: The odds of you landing on all three Light Blues (3 of 3) are an abysmal 1.0%.

However, the odds of landing on a subset of these properties are drastically higher.

The odds of landing on 1 out of 2 Browns is 33.5%.

The odds of landing on 2 out of 3. Light Blues is 11.2%. (I just increased my odds of completing a color set, without trading a set for set, by 10x)

Odds of landing on 1/3 Light Blue is 33% - 40%

(Sometimes, I'll buy 2 light blues from players for $700 or $750 each on the very first turn for a 33% - 40% odds to compete the light blue set.)

By paying $750 cash for one of those properties right now, I don't have to rely on the impossible 1% to 4.5% odds.

I only need the dice to hit the remaining properties, which happens 11% (light blues) to 33% (browns) of the time. I am using my cash to bypass the worst probabilities in the game.

2. Timing is Everything (The Psychology)

Why do I do this early, before I have the other properties? Because no competent player will ever sell you a property for cash if it completes your monopoly.

If I already have two Light Blues, nobody is giving me the third for $750. But if it's turn 5, and I have zero Light Blues or Browns, they will gladly take $750 for their single because they just see a massive cash injection. They don't realize they just handed me an 11% or 33% chance to complete the most dangerous early-game sets without needing another trade.

Also keep in mind, a competent player won't sell you a property for $750 if you can use it to swap trade with another player to complete each other's sets, so doing this early is key.

3. Eliminating Board Risk (No Set-for-Set Trades)

This is the most important part of the strategy. If you wait until the mid-game to complete a monopoly, you are almost always forced into a swap trade (e.g., "I'll give you your last Orange if you give me my last Light Blue").

I can't risk it. Giving an opponent a completed set is how you lose the game. By artificially completing my Browns or Light Blues through early cash buys and upgraded dice odds, I get my monopoly without having to arm my opponents.

4. Funding, Leverage, and Defense

Once I secure that early Brown or Light Blue, it opens up the rest of my game:

Funding: An early Brown set with cheap $50 houses carries very little risk, but it generates steady income. This funds my future property purchases or pays for another $750 Light Blue single later. It can also fund your houses on future sets if you want to make a set for set trade in the future.

Trade Leverage:

If I do need to make a swap trade later, I have cheap, developed properties backing me up. A player with a set that cost $200 to buy a house, they will not only lose money to my browns, but they will lose $100 per house they have to sell back to the bank.

The Ultimate Block (Bonus): If my early buy doesn't result in me completing the set, it still serves as a perfect defensive block. No one else can get those cheap early houses.

Offloading Liabilities: If I trade a single Green or Dark Blue to get my Brown/Light Blue, I am giving my opponent a massive liability. Even if they complete the Green set, they will never be able to afford the $200 houses to make it lethal.

TL;DR: Don't wait for a natural monopoly, and don't rely on dangerous set-for-set swap trades. Pay $750 early for a single Brown or Light Blue to upgrade your dice odds from 1% to 11% (or 4.5% to 33%). You rig the board, avoid giving your opponents sets, and secure early funding.

P.S: I've also created my own online monopoly game. There are NPCs you can play against that will attempt to make trades like the ones I described. Lmk if you are interested, and I'll link you my site

Dear Folks, I have created multiple content on Machine Learning(work in progress), and they are free. I am a data scientist and a post grad degree holder in AI/ML from IIT. To help the machine learning community with important Machine Learning Concepts, I have created multiple long form videos, and structured topicwise digestible contents structured as playlists for learning.

If you go through the first two playlists:

Introductory Machine Learning Concepts
Probability Foundations: Univariate Models

You might find helpful content, I have tried explaining with intuitions, derivations, and this is work in progress. For code implementations, scikit learn website has great content on them as well. In total they have 60+ topicwise videos so far, and I think they have the potential to help folks a lot in starting with concepts, or getting with mathematical concepts, or whether you are preparing for an AI/ML/Data job interviews etc.

When I sat for my interviews, I was grilled on my project, but majority of questions from my project tested more on foundational concepts and there know how’s.

These are FREE content on youtube. This is for the benefit of the learning community.

Link: https://youtube.com/@aayushsugandh4036?si=w5MKORU2fWzLRrAJ

Hello Folks,

Have you ever wondered why we use sigmoid function so often in Machine Learning? Although it gives us a probability, it comes from Exponential families, and this exponential family, subsumes many of the distributions, that we study in Machine Learning.

In this lecture, we understand exponential families, Directional derivatives(Gradients and Hessians), study mixture Models, and understand how domain knowledge in Probabilistic Graphical Models makes our life simpler to model joint probability densities.

Timeline breakup(in hours and minutes):
0:00-0:17 - Understanding exponential families.
0:17-0:27 - Deriving Sigmoid Function for Bernoulli.
0:27-0:48 - Understanding log partition function, convex functions and proving why positive definite of hessians imply convexity, and why convex needed?
0:48-1:04 - Directional derivates(deriving gradients and hessians)
1:04-1:26 - Maximum entropy derivation of the exponential family.
1:26-1:56 - Mixture Models(Gaussians and Bernoulli Mixture Models)
1:56-2:16 - Probabilistic Graphical Models
2:16-2:34 - Markov Chains
2:34-End - Inference and Learning, Plate Notation diagram of Gaussian Mixture Models.

If you have watched earlier of my lectures from the playlist, they will help. I try explaining as if I am a learner, to simplify complex concepts. Everything I write in whiteboard, and these are completely FREE lectures to mention.

Link: https://youtu.be/T1uTBtJ7aHU?si=rozXSTjtSqPaaYb5

I started with an existing Schrödinger style field idea, where each node carries a changing wave like state. Then I added an activation layer on top of it so the system would not just remain a continuous field, but could decide when parts of the field become visible or audible.

The system runs as a live network of moving nodes. Each node has a changing internal state influenced by its position, motion, the surrounding field, recent activity, and connections to other nodes.

Most nodes stay below the activation point. When a node crosses that point, it becomes eligible to produce sound that makes the math tangible.... Which I needed.

Instead of only watching numbers or abstract waves, the field expresses itself through motion, visual flashes, timing, notes, chords, and MIDI/audio output.

It is not a fixed loop or step sequencer. It runs continuously, and new musical events emerge as the field changes.

There is also a sound layer between raw activations and the final output. It smooths timing, limits density, and controls how many keys can overlap at once, so the result stays musical instead of every active node firing immediately.

The goal was to test whether adding an activation layer on top of a Schrödinger-style field would produce structured behaviour that could be seen and heard. The result is a live system where field changes become motion, light, and music

I just wanted to see when probability got "written" into history. Now i guess I can.

I coded a double-slit style simulation with a selection layer added on top.

The first graph shows a clear interference pattern when the paths are not separated.

The second and third graphs show what happens when path information is kept or the records are left mixed together: the visible pattern smooths out.

The final graph shows the part I’m most interested in. When the records are sorted into two related groups, the hidden structure comes back as two opposite wave patterns.

The way I’m reading it is simple: probability gives the possible patterns, but selection is what decides which structure becomes visible.