Dice, decks, expected value and the mathematics of uncertainty in game design
Randomness has a bad reputation in discussions about strategy. When a player loses because of a die roll or an unlucky card draw, it is easy to conclude that luck has replaced skill. But randomness and strategy are not opposites. Randomness is a way of creating uncertainty, and strategy is largely about making good decisions when the future is uncertain.
The important design question is therefore not whether a game contains luck. It is what the random system is allowed to decide. Does it make the decision for the player, or does it create a situation the player must interpret? Does each random event begin from the same probability distribution, or does the history of previous events change what can happen next? Can players estimate the risk, improve their odds, or choose how much variance they are willing to accept?
These questions lead directly into mathematics. Fortunately, the mathematics needed to understand most tabletop randomness is not advanced. Basic probability, dependent and independent events, expected value and a practical understanding of variance are enough to explain why a die, a deck and a bag of tokens can produce very different games even when all three are described casually as ‘random.’
Randomness can make the decision, or it can create the decision
Games have used randomizers for thousands of years. Ancient games used marked sticks, knucklebones and dice; later games added shuffled cards, spinners, roulette wheels, bags of tiles and increasingly elaborate combinations of these tools. The technology changes, but the design purpose is usually similar: the system needs a way to produce a future that is not completely known in advance.
That uncertainty can play very different roles. In Snakes and Ladders, the die effectively makes the important decision. The player rolls and the system determines how far the piece moves. There is uncertainty, but very little strategy for the uncertainty to interact with.
In Catan, a die roll also changes the state of the game, but it does not normally tell players what to do. It determines which resources enter the economy. The players then decide what to build, what to trade, what to hold and how to respond to the new situation. Ticket to Ride and Splendor use shuffled cards in a similar way: randomness helps create the problem, while the player still owns the decision.
This is a useful distinction for design. Randomness is usually more interesting when it generates conditions that require judgment than when it simply produces the final answer. A random setup, a changing market, an uncertain enemy action or a variable reward can make repeated plays diverge without removing agency. The player is not asked to predict the future perfectly; they are asked to make a good decision despite not knowing it.
A random starting point is not the same as a random outcome
Randomness can also be concentrated at different moments in the experience. Patchwork randomizes the order of the patches around the board, but once play begins the available information is largely visible. Splendor shuffles its development cards, yet players make decisions from a public market. In Alibi, participants do not choose which character they receive, but the random or assigned starting role does not dictate what they will say, whom they will trust or how they will pursue their objectives.
This matters because a system can use randomness to create variety without repeatedly interrupting player control. A random starting state says, in effect, ‘Here is today’s problem.’ From that point onward, the experience may be highly strategic.
It also helps explain why the phrase ‘this game is too random’ is often too vague to be useful. Two games can contain the same number of dice rolls and still use randomness very differently. To evaluate the design, we need to ask where uncertainty enters the system, what it changes, and what decisions remain after it arrives.
Dice have no memory
A standard six-sided die is one of the cleanest examples of independent probability. Assuming the die is fair, each face has a probability of 1/6 on every roll. Previous rolls do not alter the next one.
P(rolling a 6 on the next roll) = 1/6 ≈ 16.7%
If I roll five times without seeing a 6, the probability of a 6 on the sixth roll is still 1/6. The die does not know that I have been unlucky. Believing that a 6 has become ‘due’ is the classic gambler’s fallacy.
This does not mean that a long sequence tells us nothing. We can calculate the probability of seeing at least one 6 across several rolls. It is easier to calculate the opposite event first: the probability of never rolling a 6.
P(at least one 6 in six rolls) = 1 – (5/6)^6 ≈ 66.5%
There is no contradiction between the two statements. Before the six-roll sequence begins, there is about a 66.5% chance that the sequence will contain at least one 6. But after five failures have already happened, the sixth roll is still a fresh independent event with a 16.7% chance of success.
For game design, this means that dice can produce long unlucky or lucky streaks without correcting themselves. That can be desirable when the game wants volatility and uncertainty, but it can also create frustration if success depends too heavily on a specific result appearing.
A deck remembers
A shuffled deck behaves differently when drawn cards are not immediately returned. Each draw changes the composition of what remains, so later probabilities depend on earlier results.
Imagine a ten-card deck containing one success card and nine failure cards. Before the first draw:
P(success on the first draw) = 1/10 = 10%
If the first card is a failure and stays out of the deck, nine cards remain and the success card is still among them:
P(success next | first draw was a failure) = 1/9 ≈ 11.1%
After three failures have been removed, seven cards remain:
P(success next | first three draws were failures) = 1/7 ≈ 14.3%
The failures have not made the player ‘luckier’ in a mystical sense. They have changed the physical system. The probability increased because the denominator became smaller while the success card remained.
This is dependent probability in a form that players can often feel without calculating it. If a rare card has not appeared for a long time, players may reasonably conclude that it is becoming more likely to appear soon, provided they know the deck composition and cards are not being replaced. That intuition would be wrong for a die but correct for this deck.
| System | After a failure | What changes? |
| Fair D6 | Next success is still 1/6 | Nothing: rolls are independent |
| 10-card deck, no replacement | 1/10 → 1/9 → 1/8… | Remaining deck composition |
| Bag without replacement | Probabilities shift after each draw | Remaining token composition |
Decks can create self-correcting randomness
This difference has major design consequences. With a die, a player can theoretically continue missing forever. With a finite deck or bag, removing failures can make a desired result increasingly likely, and eventually certain if the system is exhausted far enough.
That gives designers a way to create randomness with memory. A bag can contain several bad results and a few good ones; every bad result drawn and left out improves the future distribution. A deck can guarantee that a particular event occurs once somewhere within a known range. The experience remains uncertain, but the uncertainty has boundaries.
This is one reason decks and bags can feel fairer than repeated independent rolls even when their average outcomes are similar. The system itself can absorb streaks. It does not need an explicit catch-up rule because its changing composition already modifies future odds.
Of course, reshuffling resets that memory. Returning every card or token after each draw makes the system behave more like repeated die rolls. The physical component is not what determines the mathematics; replacement rules do.
Expected value tells us the average, not the experience
Probability becomes even more useful when a random event can produce several different values. Expected value, often written E(X), is the probability-weighted average outcome we would expect over many repetitions.
For a fair six-sided die:
E(D6) = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5
This does not mean that a player can roll 3.5. It means that across a very large number of rolls, the average result tends toward 3.5.
Suppose a ten-card combat deck contains nine cards with power 9 and one card with power 4. Its expected power is:
E(X) = (9×9 + 1×4) / 10 = 85/10 = 8.5
If the player permanently removes the power-4 card, every remaining card has power 9, so the expected value becomes 9. Deck building can therefore be understood mathematically as editing a probability distribution. Adding a strong card, removing a weak card or duplicating a useful effect changes not only what the deck contains but the likelihood of future outcomes.
Expected value is useful because it lets us compare options that look very different. A guaranteed reward of 5 and a 50% chance of receiving 10 both have an expected value of 5. But they do not necessarily feel equivalent, because average value is only one property of randomness.
The same average can hide very different risks
Consider two weapons. Weapon A deals a random 1-12 damage with every value equally likely. Weapon B deals a random 5-8 damage, again with every value equally likely. Both have the same expected damage:
E(A) = (1+2+…+12)/12 = 6.5 | E(B) = (5+6+7+8)/4 = 6.5
Yet nobody would describe them as identical. Weapon A can produce spectacular hits and disastrous failures. Weapon B stays close to its average. The difference is the spread of the possible outcomes.
Mathematicians describe that spread using variance and standard deviation. A full derivation is unnecessary for most design work, but the intuition is essential: higher variance means results are more dispersed around the average; lower variance means results are more consistent.
For these two examples, Weapon A has a standard deviation of about 3.45 damage, while Weapon B is about 1.12. The averages are identical, but Weapon A is roughly three times as volatile around that average.
That volatility is a design choice. A high-variance option may suit a player who is behind and needs a dramatic swing, while a player who is already ahead may prefer reliability. Once players can choose between those risk profiles, randomness itself becomes part of strategy rather than an interruption of it.
Probability is most interesting when players can act on it
A probability calculation becomes meaningful in a game when it changes a decision. If I need a 6 and there is nothing I can do except roll, knowing that my chance is 1/6 may explain the result but does not create much strategy. If I can choose between a safe action with a modest reward and a risky action with a larger expected payoff, probability becomes part of the decision space.
The same is true when players can manipulate a deck, count remaining cards, spend resources to reroll, improve a target number, draw additional options, or decide whether to stop while they are ahead. These mechanisms allow players to manage uncertainty instead of merely receiving it.
Good strategy does not require certainty. In fact, certainty often reduces the need for strategy. A plan becomes interesting when it has to survive several plausible futures. The player asks not only ‘What happens if this works?’ but ‘How badly am I hurt if it does not?’, ‘What other outcomes can I tolerate?’, and ‘Can I improve my position before taking the risk?’
This is why randomness can support mastery. Experienced players may not control the random result, but they become better at understanding distributions, preserving flexibility, recognizing when the odds have shifted and choosing risks appropriate to the current state of the game.
What kind of uncertainty do you want the player to manage?
When choosing a random mechanism, the first question should not be ‘dice or cards?’ It should be what kind of uncertainty the game needs.
Independent rolls are useful when every event should begin fresh and previous outcomes should not constrain the next one. Decks and bags without replacement are useful when the past should change the future, when players should be able to infer something from what has already appeared, or when the design wants to limit extreme streaks. Random setup can create replayability while leaving the rest of the game highly deterministic. High variance can create dramatic swings; low variance can make planning more reliable.
Expected value helps us understand the average strength of an option. Variance helps us understand how unreliable that average may be. Conditional probability tells us when new information should change our estimate. None of these concepts tells us automatically what the design should do, but together they give us a much better language for deciding what kind of uncertainty we are creating.
Randomness is therefore not the opposite of strategy. It is one of the materials from which strategic problems can be built. The designer’s job is not simply to decide how much luck a game should contain, but to decide what the random system knows, what the player knows, what previous events change, and what meaningful choices remain after uncertainty enters the game.
A die, a deck and a bag may all generate an unpredictable result. Mathematically and experientially, however, they can produce very different futures. Understanding that difference is part of understanding games.