How to always win in UNO
- Amanda Munandar
- May 5
- 4 min read
Using Probability in UNO
UNO is a game that blends luck, strategy, and a bit of psychological tactics. While chance plays a role in card draws, applying mathematics, probability, and strategic gameplay can give you a significant edge. Using math will not only win you a game of UNO, but also the following bragging rights.
The 108-card deck and the 7-card initial hand create over 27 billion possible combinations.
A strong opening hand with wilds or action cards provides a tactical advantage. The way you sequence your cards also affects your ability to win. We can model an UNO game as a probabilistic system: the deck shuffle and draw introduce randomness, while players’ choices introduce strategic complexity and conditional probabilities.
Winning at UNO is about making the most optimal decisions. If you have multiple playable cards, choosing the best one depends on expected value (EV). Playing a Wild early might change the game’s color in your favor but could be wasted if you don’t have a matching follow-up. Holding onto Draw Twos and Skips until a critical moment can give you control over opponents. Playing high-value cards early reduces the penalty if you are forced to draw later.
A Markov chain is a mathematical model that represents game states and the probability of transitioning from one state to another. In UNO, each state can be defined as the number of cards in each player’s hand. The probability of winning can be calculated by modeling state transitions (playing, drawing, or skipping turns). The best strategy is to always transition toward a lower-card state while forcing opponents into higher-card states by making them draw or skip turns.
In UNO, we can define states based on the number of cards a player has. Let’s assume a simplified model:
State S₀: The player has zero cards (they won).
State S₁: The player has one card (UNO!).
State S₂: The player has two cards.
…
State Sₙ: The player has n cards.
Each turn, the player can:
Play a card (moving to Sₖ₋₁ with probability P₁).
Draw a card (moving to Sₖ₊₁ with probability P₂).
Stay at the same state (due to a forced skip, probability P₃)
An absorbing Markov chain has certain states where, once entered, the process cannot leave. These are called absorbing states. In UNO, the absorbing state is S₀ (having 0 cards), meaning the game ends. Every other state (S₁, S₂, …, Sₙ) eventually transitions into S₀ (someone wins).
Using absorbing Markov chains, we can calculate:
The probability of winning given a certain hand size.
Expected turns to win from a given state.
The best actions to reach S₀ faster.
For example, if we model a 5-player game, we might find that if a player has 1 card left, they will win 70% of the time within 2 turns. If a player has 3 cards left, their expected number of turns to win is 5. A player with 7+ cards has a low chance of winning soon, so forcing draws on other players is the best play.
The Monte Carlo method uses random sampling to approximate complex probability problems. Instead of solving equations, we run thousands or millions of trials and analyze the outcomes. Let’s say we want to know whether it’s better to play a Wild card immediately or hold it. Theoretically, the best move depends on:
How many turns the game is expected to last.
Whether opponents have Draw Two cards.
The probability of drawing a playable card later.
Instead of solving this analytically, we can run a Monte Carlo simulation:
Simulate 100,000 games where Player A plays Wild immediately.
Simulate 100,000 games where Player A holds Wild until late-game.
Compare the win rates of each strategy.
Monte Carlo methods allow us to empirically determine the best strategy through computational brute force, rather than solving difficult equations. Of course, using algorithms can help simulate 100,000 UNO games with your friends, but The Monte Carlo method is more practical in modelling stock market trends, analysing system failures in engineering, or disease spread in medicine.
Bayesian probability is a framework for updating probabilities based on new evidence. In simpler terms, Bayesian probability allows you to update your beliefs about an event as new information is revealed. In UNO, you often need to infer hidden information based on how opponents play. Bayesian reasoning helps refine your estimates as the game progresses.
Imagine it’s late in the game, and your opponent has just one card left (UNO!). You have two cards left: a Blue 3 and a Wild. You need to decide whether to change the color using a Wild card.
You have two choices:
Play the Wild and choose Blue.
This would guarantee your last card is playable next turn.
However, if the next player has a Blue card, you might lose immediately.
Play the Blue 3 instead.
If the next player doesn’t have a Blue, you can play the Wild next turn to guarantee a win.
But if they do have a Blue, you lose.
Using decision theory, we can assign probabilities. If the opponent’s chance of holding a Blue card is 30%, then playing Wild first gives a 70% win rate. Playing Blue first only wins if they don’t have a Blue, which is also 70%. Both moves seem equal, but if the next player’s reaction suggests uncertainty, Bayesian reasoning suggests they likely don’t have Blue—in which case, keeping the Wild may be better. By analyzing decision-making probabilities, we make more optimal strategic plays in UNO.
UNO might seem like a straightforward game, but concepts from Markov Chains, Monte Carlo methods, Game Theory, and Decision Theory help explain the logic behind optimal play. These theories don't necessarily introduce new ideas—rather, they formalize decisions that players instinctively make. Everyone naturally aims to get rid of their cards as efficiently as possible, but probability and math help break down the best ways to do it, showing how these principles play a part in everyday decision-making.
References
Analyzing Probability in UNO: Strategies & Chances of winning | Course Hero. (2024, April 6). https://www.coursehero.com/file/230033718/-Analysis-of-a-Game-3pdf/
Markov Chains and stochastic stability. (Meyn & Richard). Google Books.
Markov Chains explained visually. (n.d.). Explained Visually.
The Editors of Encyclopaedia Britannica. (2025, January 17). Monte Carlo method | Simulation, Probability & Statistics. Encyclopedia Britannica.
The Editors of Encyclopaedia Britannica. (2011, September 19). Bayesian analysis | Probability Theory, Statistical Inference. Encyclopedia Britannica.
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