Loaded Dice: Bias and Applications

• Loaded dice are finite random sources with nonuniform face probabilities used to model biased outcomes and analyze entropy in various probabilistic frameworks.
• They enable efficient algorithms for extracting unbiased bits from biased outcomes, achieving asymptotic optimality and reducing computational overhead.
• Beyond probability theory, loaded dice serve as metaphors and artifacts in fields like climate science, HCI, and computer architecture for designing bias-aware systems.

Loaded dice are, in the most direct probabilistic sense, finite random sources with nonuniform face probabilities. In the cited literature, the term also appears in several extended senses: as a device for extracting or simulating randomness, as an object of inference and algebraic analysis, as a multisensory ideation artifact in HCI, as a metaphor for shifted climate probabilities, and as the name of a RowHammer mitigation mechanism that biases protection toward repeatedly sampled rows (Zhou et al., 2012, Kurze, 2022, Hansen et al., 2012, Woo et al., 17 May 2026).

1. Formal probabilistic definitions

A loaded $m$-sided die is defined as a source that on each roll produces a symbol in the alphabet $\{1,2,\ldots,m\}$, with

$$P[\text{face }i] = p_i,\qquad i=1,\ldots,m,\qquad \sum_{i=1}^m p_i = 1.$$

Its entropy is

$H(p_1,\ldots,p_m) = - \sum_{i=1}^m p_i \log_2 p_i.$

This formulation places loaded dice alongside biased coins as basic discrete sources, with the biased 2-face coin recovered as the case $m=2$ (Zhou et al., 2012).

A second formalization considers repeated rolls. For a $q$-sided die with weights $p_0,\ldots,p_{q-1}$, let $x_1,x_2,\ldots$ be i.i.d. outcomes in $\{0,1,\ldots,q-1\}$, and define

$$X = \sum_{i=1}^{\infty} x_i q^{-i},\qquad X\in[0,1].$$

The induced cumulative distribution function is

$\{1,2,\ldots,m\}$0

This construction shifts the focus from single-roll probabilities to the measure induced on $\{1,2,\ldots,m\}$1 by an infinite $\{1,2,\ldots,m\}$2-ary expansion (Pfeffer et al., 2023).

Independently rolled loaded dice also define product probability spaces. For two six-sided dice with face probabilities $\{1,2,\ldots,m\}$3 and $\{1,2,\ldots,m\}$4, independence gives $\{1,2,\ldots,m\}$5, and the distribution of the total $\{1,2,\ldots,m\}$6 is

$\{1,2,\ldots,m\}$7

This total-law perspective underlies inverse questions about whether factor distributions are determined by the distribution of sums (Morrison et al., 2013).

2. Extraction of unbiased bits from biased dice

A central algorithmic problem is to convert biased $\{1,2,\ldots,m\}$8-ary outcomes into independent unbiased bits. One universal scheme begins with any fixed-length coin-to-bits algorithm $\{1,2,\ldots,m\}$9 that extracts unbiased bits from biased 2-face coins, and lifts it to an algorithm $$P[\text{face }i] = p_i,\qquad i=1,\ldots,m,\qquad \sum_{i=1}^m p_i = 1.$$0 for $$P[\text{face }i] = p_i,\qquad i=1,\ldots,m,\qquad \sum_{i=1}^m p_i = 1.$$1-sided dice. The construction uses a binarization tree of depth $$P[\text{face }i] = p_i,\qquad i=1,\ldots,m,\qquad \sum_{i=1}^m p_i = 1.$$2: each die face is represented by a binary string of length $$P[\text{face }i] = p_i,\qquad i=1,\ldots,m,\qquad \sum_{i=1}^m p_i = 1.$$3, and as a die-outcome sequence $$P[\text{face }i] = p_i,\qquad i=1,\ldots,m,\qquad \sum_{i=1}^m p_i = 1.$$4 is read, bits are routed to buffers $$P[\text{face }i] = p_i,\qquad i=1,\ldots,m,\qquad \sum_{i=1}^m p_i = 1.$$5 attached to tree nodes $$P[\text{face }i] = p_i,\qquad i=1,\ldots,m,\qquad \sum_{i=1}^m p_i = 1.$$6. The derived extractor is

$$P[\text{face }i] = p_i,\qquad i=1,\ldots,m,\qquad \sum_{i=1}^m p_i = 1.$$7

The key structural fact is invertibility: from the collection $$P[\text{face }i] = p_i,\qquad i=1,\ldots,m,\qquad \sum_{i=1}^m p_i = 1.$$8 one can uniquely reconstruct $$P[\text{face }i] = p_i,\qquad i=1,\ldots,m,\qquad \sum_{i=1}^m p_i = 1.$$9. The correctness theorem then states that if $H(p_1,\ldots,p_m) = - \sum_{i=1}^m p_i \log_2 p_i.$0 produces independent unbiased bits, so does the induced $H(p_1,\ldots,p_m) = - \sum_{i=1}^m p_i \log_2 p_i.$1 for an $H(p_1,\ldots,p_m) = - \sum_{i=1}^m p_i \log_2 p_i.$2-sided die with arbitrary biases $H(p_1,\ldots,p_m) = - \sum_{i=1}^m p_i \log_2 p_i.$3 (Zhou et al., 2012).

The same framework yields asymptotic optimality. If the coin algorithm $H(p_1,\ldots,p_m) = - \sum_{i=1}^m p_i \log_2 p_i.$4 is asymptotically optimal, then for $H(p_1,\ldots,p_m) = - \sum_{i=1}^m p_i \log_2 p_i.$5,

$H(p_1,\ldots,p_m) = - \sum_{i=1}^m p_i \log_2 p_i.$6

Equivalently, if $H(p_1,\ldots,p_m) = - \sum_{i=1}^m p_i \log_2 p_i.$7 is the number of die rolls needed to generate a fixed $H(p_1,\ldots,p_m) = - \sum_{i=1}^m p_i \log_2 p_i.$8 bits, then

$H(p_1,\ldots,p_m) = - \sum_{i=1}^m p_i \log_2 p_i.$9

The reported implementation cost is $m=2$0, with total buffer length $m=2$1 and up to $m=2$2 buffers in the straightforward tree representation (Zhou et al., 2012).

The same paper also studies prescribed-length extraction. Its iterative variable-to-fixed-length scheme $m=2$3 is stated for biased coins rather than general dice: each round produces $m=2$4 bits with $m=2$5, produces $m=2$6 with probability at least $m=2$7, and reaches exactly $m=2$8 bits by invoking $m=2$9 on fresh tosses whenever $q$0. The expected number of iterations is at most $q$1, and the total toss complexity satisfies

$q$2

A plausible implication is that the die-to-bits transformation and the fixed-length coin extractor can be composed when exact output length is operationally required (Zhou et al., 2012).

3. Exact discrete sampling and die-to-die factories

Loaded dice also arise as target distributions to be sampled exactly from unbiased random bits. The Fast Loaded Dice Roller (FLDR) addresses the problem of generating a random integer from a discrete distribution $q$3, equivalently from positive integer weights $q$4 summing to $q$5. FLDR extends the weights with $q$6, where $q$7, thereby forming a dyadic proposal distribution $q$8 over $q$9 outcomes. It then simulates $p_0,\ldots,p_{q-1}$0 by an implicit entropy-optimal DDG tree of depth $p_0,\ldots,p_{q-1}$1, rejects the extra outcome $p_0,\ldots,p_{q-1}$2, and stores the sampler in two arrays: $p_0,\ldots,p_{q-1}$3, the number of leaves at each level, and $p_0,\ldots,p_{q-1}$4, the labels of those leaves in sorted order (Saad et al., 2020).

The reported preprocessing cost is $p_0,\ldots,p_{q-1}$5 time and $p_0,\ldots,p_{q-1}$6 space. Sampling uses $p_0,\ldots,p_{q-1}$7 integer operations per consumed bit, and the main theoretical guarantee is

$p_0,\ldots,p_{q-1}$8

The same source reports that FLDR is $p_0,\ldots,p_{q-1}$9–$x_1,x_2,\ldots$0 faster in both preprocessing and sampling than multiple baseline algorithms, including alias and interval samplers, and uses up to $x_1,x_2,\ldots$1 less space than a full entropy-optimal sampler (Saad et al., 2020).

A more general “dice enterprise” problem begins with an $x_1,x_2,\ldots$2-sided die of unknown face-probabilities $x_1,x_2,\ldots$3 and asks for simulation of a $x_1,x_2,\ldots$4-sided die with probabilities $x_1,x_2,\ldots$5, where $x_1,x_2,\ldots$6 is a rational map between open probability simplices. The construction rewrites $x_1,x_2,\ldots$7 using a common positive denominator $x_1,x_2,\ldots$8 and homogeneous positive polynomials, producing a fine, connected multivariate ladder

$x_1,x_2,\ldots$9

A finite-state Markov chain is then built with stationary distribution $\{0,1,\ldots,q-1\}$0; exact simulation is obtained via Coupling from the Past, with a monotone two-chain specialization in the Bernoulli-factory case $\{0,1,\ldots,q-1\}$1. For the Bernoulli-factory case, the number of tosses has exponential tails and its expected value can be bounded uniformly in $\{0,1,\ldots,q-1\}$2 (Morina et al., 2019).

These two lines of work are complementary. FLDR starts from unbiased bits and produces a loaded-dice sample exactly; Dice Enterprise starts from a loaded die with unknown parameters and constructs another die whose law is a rational transform of the original law. This suggests a broad algorithmic view in which “loaded dice” are both sources and targets of exact stochastic compilation (Saad et al., 2020, Morina et al., 2019).

4. Infinite-roll distributions and singularity

When a loaded die is rolled infinitely often and interpreted through a $\{0,1,\ldots,q-1\}$3-ary expansion, the induced distribution on $\{0,1,\ldots,q-1\}$4 has a recursive structure. If $\{0,1,\ldots,q-1\}$5, then for $\{0,1,\ldots,q-1\}$6,

$\{0,1,\ldots,q-1\}$7

For example, on $\{0,1,\ldots,q-1\}$8, $\{0,1,\ldots,q-1\}$9, while on $$X = \sum_{i=1}^{\infty} x_i q^{-i},\qquad X\in[0,1].$$0,

$$X = \sum_{i=1}^{\infty} x_i q^{-i},\qquad X\in[0,1].$$1

Iterating the recursion over finer $$X = \sum_{i=1}^{\infty} x_i q^{-i},\qquad X\in[0,1].$$2-ary intervals yields a continuous piecewise-linear construction at every finite stage (Pfeffer et al., 2023).

The principal qualitative result is singularity. If some $$X = \sum_{i=1}^{\infty} x_i q^{-i},\qquad X\in[0,1].$$3, then the induced measure $$X = \sum_{i=1}^{\infty} x_i q^{-i},\qquad X\in[0,1].$$4 is singular with respect to Lebesgue measure, and the CDF satisfies $$X = \sum_{i=1}^{\infty} x_i q^{-i},\qquad X\in[0,1].$$5 Lebesgue-a.e. The argument compares $$X = \sum_{i=1}^{\infty} x_i q^{-i},\qquad X\in[0,1].$$6-typical digit frequencies, which converge to $$X = \sum_{i=1}^{\infty} x_i q^{-i},\qquad X\in[0,1].$$7 by the strong law of large numbers, with Lebesgue-typical frequencies, which equal $$X = \sum_{i=1}^{\infty} x_i q^{-i},\qquad X\in[0,1].$$8 by normal number theory. In contrast, in the uniform case $$X = \sum_{i=1}^{\infty} x_i q^{-i},\qquad X\in[0,1].$$9 for all $\{1,2,\ldots,m\}$00, the recursion gives $\{1,2,\ldots,m\}$01, so the law is absolutely continuous (Pfeffer et al., 2023).

The paper develops two quantitative comparisons with the uniform CDF $\{1,2,\ldots,m\}$02. The first is the sup-norm $\{1,2,\ldots,m\}$03, using a contraction operator $\{1,2,\ldots,m\}$04 on $\{1,2,\ldots,m\}$05 with Lipschitz constant $\{1,2,\ldots,m\}$06. The second is arclength: $\{1,2,\ldots,m\}$07 For any continuous, increasing $\{1,2,\ldots,m\}$08 with $\{1,2,\ldots,m\}$09 and $\{1,2,\ldots,m\}$10, the cited theorem states that $\{1,2,\ldots,m\}$11 if and only if $\{1,2,\ldots,m\}$12 is singular, while $\{1,2,\ldots,m\}$13. Hence $\{1,2,\ldots,m\}$14 for any unfair die and $\{1,2,\ldots,m\}$15 only in the fair case (Pfeffer et al., 2023).

In the special case $\{1,2,\ldots,m\}$16, the model reduces to a biased coin with

$\{1,2,\ldots,m\}$17

If $\{1,2,\ldots,m\}$18, then $\{1,2,\ldots,m\}$19; otherwise the limit law is the classical singular Bernoulli-convolution described in the paper as a devil’s staircase (Pfeffer et al., 2023).

5. Inference, maximum entropy, and inverse problems

A different class of loaded-dice problems asks what can be inferred about unknown face probabilities from partial information. In the Brandeis Dice Problem, the observable constraint is a known mean face value

$\{1,2,\ldots,m\}$20

Jaynes’ Maximum Entropy approach infers probabilities $\{1,2,\ldots,m\}$21 by maximizing Shannon entropy

$\{1,2,\ldots,m\}$22

subject to $\{1,2,\ldots,m\}$23 and $\{1,2,\ldots,m\}$24. The resulting canonical-form solution is

$\{1,2,\ldots,m\}$25

with $\{1,2,\ldots,m\}$26 fixed by

$\{1,2,\ldots,m\}$27

The same paper presents a Bayesian alternative at a deeper level of description, introducing latent chances $\{1,2,\ldots,m\}$28 with a uniform prior on the simplex and obtaining, in the $\{1,2,\ldots,m\}$29 limit with only the average observed, a micro-canonical posterior

$\{1,2,\ldots,m\}$30

The next-toss prediction is then the posterior mean $\{1,2,\ldots,m\}$31 (Enk, 2014).

The significance of this comparison is not merely conceptual. The cited analysis states that, for finite $\{1,2,\ldots,m\}$32, the canonical-MaxEnt probabilities and the microcanonical posterior means are not equivalent, but become equivalent in the analogue of the thermodynamic limit $\{1,2,\ldots,m\}$33. It also reports error bars: for $\{1,2,\ldots,m\}$34 and $\{1,2,\ldots,m\}$35, both methods give $\{1,2,\ldots,m\}$36 of order $\{1,2,\ldots,m\}$37–$\{1,2,\ldots,m\}$38, and for finite $\{1,2,\ldots,m\}$39 the two predictions can differ by $\{1,2,\ldots,m\}$40 or more (Enk, 2014).

A related inverse problem concerns identifiability from total distributions. For two independent six-sided dice with face probabilities $\{1,2,\ldots,m\}$41 and $\{1,2,\ldots,m\}$42, one may ask whether there exist non-uniform probabilities such that the total-law matches the fair-craps distribution

$\{1,2,\ldots,m\}$43

Three proofs are summarized for the negative answer. A Gröbner-basis computation shows that the only real nonnegative solution is $\{1,2,\ldots,m\}$44. Over $\{1,2,\ldots,m\}$45, the solution set consists of exactly 51 points, but only the fair-dice point is nonnegative real. A generating-function proof factors

$\{1,2,\ldots,m\}$46

and concludes that the only admissible split into degree-5 factors with nonnegative coefficients is the trivial fair one. The same source notes that, by contrast, explicit positive-coefficient counterexamples appear for larger orders, with examples starting at order $\{1,2,\ldots,m\}$47 (Morrison et al., 2013).

6. Loaded Dice in interaction design

In HCI and design research, “Loaded Dice” denotes a multisensory ideation tool for connected devices. The system consists of two wireless, battery-powered cubes of approximately $\{1,2,\ldots,m\}$48 edge length, each with six functional faces, where the “top” face is always the active face. One cube is a sensor die with temperature sensor, ambient light sensor, microphone, 3-axis movement sensor, potentiometer, and ultrasonic distance sensor. The other is an actuator die with vibration motor, heating surface, LED bar-graph, loudspeaker, power LEDs, and miniature fan (Kurze, 2022).

The cubes are embedded in workshops through an extended interaction vocabulary adapted from earlier interaction-vocabulary work. The paper lists pairs such as approximate—precise, casual—attention-grabbing, direct—mediated, friendly—angry, graded—binary, incidental—targeted, instant—delayed, objective—poetic, powerful—gentle, private—public, slow—fast, and soft—angry. In the “Cards’n’Dice” method, workshops begin with an introduction of 10 minutes, proceed through 4–6 ideation rounds of 15 minutes each, and end with 20 minutes of reflection. Each round has teams pick one sensor cube face and one actuator cube face, draw 1–2 interaction-quality cards, sketch or role-play a scenario, rapidly “micro-prototype” it by holding the cubes together, and then present insights and refine (Kurze, 2022).

The stated goals are to encourage rapid, playful exploration of 36 possible sensor–actuator combinations, stimulate thinking beyond screens and buttons into temperature, motion, sound, and prospectively smell, and surface latent preferences such as “private” versus “public” or “incidental” versus “targeted.” The paper additionally outlines a smell extension: a small cartridge or absorbent pad soaked with scented oil is placed inside the actuator cube, the existing Peltier heater is used to volatilize the odorant, and the mini-fan directs the scented airflow. The described control strategy uses discrete Peltier temperature levels such as $\{1,2,\ldots,m\}$49, $\{1,2,\ldots,m\}$50, and $\{1,2,\ldots,m\}$51, together with PWM-controlled fan duty cycle to set smell strength (Kurze, 2022).

The evaluation claims in this line of work are explicitly limited. The paper reports no formal user-study metrics, but emphasizes repeated observations across multiple co-design sessions: smell broadened the space of envisioned scenarios, generated ideas around non-visual and non-audible channels for intimate or ambient communication, and increased metaphorical uses such as “warmth as digital hug.” This suggests that, in this context, “loaded” refers not to probabilistic bias but to preconfigured multisensory affordances that deliberately bias ideation toward unconventional IoT experiences (Kurze, 2022).

7. Metaphorical and architectural extensions

In climate science communication, “loaded dice” is a metaphor for the shifted and broadened distribution of seasonal temperature anomalies. Using the 1951–1980 climatology as baseline, a six-sided climate die is defined with two blue faces for “cold,” two white faces for “average,” and two red faces for “hot,” corresponding to terciles under a standard normal distribution. The tercile cut points are at $\{1,2,\ldots,m\}$52, and the normalized anomaly is

$\{1,2,\ldots,m\}$53

The cited paper reports that “hot” summers $\{1,2,\ldots,m\}$54 rose from approximately $\{1,2,\ldots,m\}$55 in 1951–80 to approximately $\{1,2,\ldots,m\}$56 in 2001–11; “very hot” $\{1,2,\ldots,m\}$57 rose from approximately $\{1,2,\ldots,m\}$58 to approximately $\{1,2,\ldots,m\}$59 of land; and “extremely hot” $\{1,2,\ldots,m\}$60 rose from approximately $\{1,2,\ldots,m\}$61–$\{1,2,\ldots,m\}$62 in 1951–80 to approximately $\{1,2,\ldots,m\}$63 in 2006–11, compared with the baseline Gaussian expectation of $\{1,2,\ldots,m\}$64 beyond $\{1,2,\ldots,m\}$65 (Hansen et al., 2012).

The metaphor is designed to make altered odds perceptible without erasing variability. The same source argues that fixing the baseline at 1951–1980, rather than continuously updating “normal,” prevents the dice from appearing perpetually fair while the underlying distribution shifts. It further states, with respect to the 2010 Moscow heatwave and the 2011 Texas/Oklahoma heat and drought, that such $\{1,2,\ldots,m\}$66 anomalies were “effectively impossible” in the absence of global warming because their probability under the old distribution was vanishingly small (Hansen et al., 2012).

In computer architecture, “Loaded Dice” is also the subtitle of PrISM, a scalable probabilistic RowHammer defense. The motivating problem is the non-selection problem in fixed-rate probabilistic defenses such as MINT: at low thresholds $\{1,2,\ldots,m\}$67–$\{1,2,\ldots,m\}$68, a heavily hammered row can repeatedly escape sampling across mitigation windows. PrISM addresses this by sampling $\{1,2,\ldots,m\}$69 activation slots per window of size $\{1,2,\ldots,m\}$70, storing sampled-but-unmitigated rows in a per-bank Sampled History Queue (SHQ) across the previous $\{1,2,\ldots,m\}$71 windows, and issuing an extra mitigation through Alert Back-Off whenever a newly sampled row reappears in the SHQ. The paper gives

$\{1,2,\ldots,m\}$72

and interprets history intersections as a mechanism that “loads” mitigation toward persistent aggressors (Woo et al., 17 May 2026).

The reported overheads are explicit. At threshold $\{1,2,\ldots,m\}$73, PrISM incurs a $\{1,2,\ldots,m\}$74 average slowdown compared to $\{1,2,\ldots,m\}$75 for PRAC; at threshold $\{1,2,\ldots,m\}$76, it reduces average slowdown from $\{1,2,\ldots,m\}$77 for MINT to $\{1,2,\ldots,m\}$78, a $\{1,2,\ldots,m\}$79 reduction. The required storage is modest: at $\{1,2,\ldots,m\}$80, the SHQ has 246 entries, and the paper states a maximum of $\{1,2,\ldots,m\}$81 of SRAM per bank for the SHQ and less than $\{1,2,\ldots,m\}$82 per bank overall, with no DRAM timing-parameter changes (Woo et al., 17 May 2026).

Across these examples, the phrase “loaded dice” retains the core idea of biased outcomes, but the object being biased differs. In climate science, the biased object is a public probability metaphor for seasonal anomalies; in PrISM, it is the mitigation process itself, made history-sensitive so that repeated row activity is disproportionately likely to trigger protection (Hansen et al., 2012, Woo et al., 17 May 2026).