This card game, besides being easy for socializing people at 'bonding' events, UNO is also easy to memorize and play. UNO is often considered a 'children's game' that is no more difficult than other card games like poker, bridge, or other variations of playing card games. Unlike poker, playing UNO doesn't require memorizing many rules.
Before I discuss further about the probabilities that might be encountered when playing UNO, I will discuss the general rules of this game, especially about the composition of one UNO deck. One UNO deck consists of 108 cards with details: 72 cards numbered 1-9 (2 of each color), 4 zero cards (1 of each color), 8 +2 cards (2 of each color), 8 reverse cards (2 of each color), 8 skip cards (2 of each color), 4 neutral black cards, and 4 neutral black +4 cards.
The core rule of the UNO game is whoever finishes their cards fastest is the winner. I assume readers already know how to play UNO (if not, Google it, there's plenty of info). The game starts by dealing cards to each player, with each player getting 7 cards that have been shuffled beforehand. Yes, however random becomes an important assumption when explaining everything with probability.
Total number of all possible card combinations
All possibilities of the seven cards we get are called the universal set in set theory. In sampling techniques in statistics, we also know the concept of all possible samples as the universe of possible sample combinations. We can calculate the total number of all possible combinations of seven cards with the combination formula as follows:
n(S) = nCr = (108)C(7) = 27,883,218,170.
Fantastic! Therefore, I won't mention the combinations, hehe.
Probability of getting a combination with no action cards (most unlucky probability)
Based on the initial rules, action cards consist of 0, +2, +4, black, skip, and reverse totaling 36 cards. Before calculating the probability of getting this most unlucky combination, let's first calculate the number of possible combinations without action cards. The total number of combinations of seven cards without action cards is the combination of 7 from the number of non-action cards, which is 72 cards. Mathematically it can be written as n(A) = 72C7 = 1,473,109,704 combinations.
The probability of getting a combination with no action cards can be calculated using the probability formula as follows:
P(A) = n(A)/n(S) = 1,473,109,704/27,883,218,170 = 0.5283 = 52.83%.
Hmm, quite significant probability, exceeding half. From this we can also know the probability of getting a combination with action cards (whether one or more) is as follows:
P(A') = 1 – P(A) = 1 – 0.5283 = 0.4717 = 47.17%
What can be interpreted from this probability calculation is that the probability of getting at least 1 action card versus getting no action cards at all is almost the same (only about 5% apart). Using the concept of expected frequency (expected value), for example if we play UNO 100 times, then our expectation of getting at least one action card is 47 times, and the expectation of not getting any action cards at all is 53 times. Yes, once again we call this "expectation," expectations don't always have to match reality, right?
Probability of getting all seven cards as action cards (most lucky probability)
After calculating the most unlucky probability in playing UNO, it wouldn't be complete if we didn't also calculate the luckiest probability. Someone is luckiest, I assume, when we get a combination where all seven cards are action cards (whether zero, +2, +4, black, skip, or reverse). As we calculated before, there are 36 action cards out of 108 total cards. So the number of combinations where all seven cards are action cards is n(B) = 36C7 = 8,347,680 card combinations. The probability of getting a combination where all seven cards are action cards can be calculated as follows:
P(B) = n(B)/n(S) = 8,347,680/27,883,218,170 = 0.00030 = 0.030%.
Hmm, very little, even though I wanted to break it down further. From that probability, our expectation of getting "one" combination where everything is action cards is after playing 3,333 times. Want to prove it? Hehe, you'd get tired.
Probability of all seven cards being black (automatic win version 1)
In the UNO game, in my opinion, someone is considered very lucky if their initial seven cards are all black, because they can immediately win in at least two turns without any obstacles except if opponents play skip, reverse, and +2 against them which could cancel the victory.
The condition where all seven cards are black is a derivative of the condition where all seven cards are action cards. The probability of all seven cards being black plus the probability of having at least one non-black action card equals the probability of all seven cards being action cards. Mathematically it can be written as follows:
B1 = all seven cards are black B2 = at least one non-black action card exists
B = all seven cards are action cards P(B1) + P(B2) = P(B)
The number of black cards in one UNO deck is 8 cards consisting of 4 neutral black cards and 4 black +4 cards. Like in the previous step, we must calculate the combination where all seven cards are black, which is 8C7 = 8 card combinations. So, from here the probability can be calculated as follows:
P(B1) = n(B1)/n(S) = 8/27,883,218,170 = 2.86 x 10^(-10)
Using the concept of expected frequency, our expectation to get one combination where all cards are black requires playing 3,496,503,497 times. Okay, this is getting ridiculous, only the luckiest people could get this card combination.
Probability of getting (at least one) black card
If the probability of getting all seven cards as black cards is not easily obtained, don't worry, we can still win if we get any number of black cards. Black cards are indeed often considered sacred because they can beat various colored cards (except +2).
The condition of getting at least one black card is the complement of getting a combination where all cards are colored (no black cards at all). So, to simplify the calculation, we should start by calculating the probability of getting a combination with no black cards at all.
As we know, there are 100 colored cards consisting of 76 cards numbered 0-9, 8 reverse, 8 skip, and 8 +2 cards. So, the number of combinations is: 100C7 = 16,007,560,800 combinations. The probability is as follows:
P(C) = n(C)/n(S) = 16,007,560,800/27,883,218,170 = 0.5741 = 57.41%
So the probability of getting (at least one) black card is P(C') = 1-P(C) = 1-0.5741 = 0.4259 = 42.59%. Quite significant, right? In 100 games, our expectation of getting at least one black card is 42 to 43 times. Basically, this probability becomes more accurate if we keep experimenting, or the more often we play.
Probability of getting all seven cards with the same number/symbol (automatic win version 2)
A combination of seven cards with the same number can win the game quickly (variant UNO rules state that cards with the same number can be played simultaneously). Calculating this combination requires the concept of union, which is adding all possible combinations.
If we want to calculate the probability of seven cards having the same number, we must calculate the combination of seven cards numbered 1, or 2, 3, and so on. Mathematically it can be formulated as:
N(1) + N(2) + … + N(9) + N(reverse) + N(+2) + N(skip) = n(Q)
For numbered cards 1-9, each has 8 cards in the deck (2 per color), so the number of combinations for each number is 8C7 = 8 combinations.
For special cards (reverse, +2, skip), each also has 8 cards in the deck, so the number of combinations is also 8C7 = 8 combinations each.
Therefore: n(Q) = 9×8 + 3×8 = 72 + 24 = 96 combinations.
The probability is:
P(Q) = n(Q)/n(S) = 96/27,883,218,170 = 3.44 × 10^(-9) = 0.000000344%
This is incredibly rare! You'd need to play about 290 million times to expect this once.
What makes UNO fascinating from a mathematical perspective is how it balances randomness with strategy. While perfect hands are nearly impossible, the game provides enough variability in card distribution to keep games interesting while maintaining overall fairness.
The lesson here is that even in games that seem purely based on luck, mathematical analysis can reveal the underlying structure and help us understand why certain games remain engaging over time. UNO's enduring popularity isn't just about simple rules—it's about well-designed probability distributions that create excitement without making outcomes entirely random.
Understanding these probabilities doesn't guarantee winning, but it does help appreciate the mathematical elegance hidden within this seemingly simple card game.