About

Decision fatigue is a cognitive bottleneck in high-stakes environments. This tool provides a deterministic chaos engine to resolve binary disputes, test probability models, or conduct impartial sampling. Unlike standard pseudo-random generators that simply output a text string, this application simulates the physical dynamics of a Bernoulli trial - the mathematical definition of a coin toss.

Accuracy is paramount in statistical modelling. Browser-based Math.random() implementations vary in entropy quality. This tool utilizes a cryptographic-strength randomizer to ensure uniform distribution, critical for developers testing binomial algorithms or Game Masters requiring truly unbiased outcomes. For educational and experimental purposes, the engine supports "Weighted Biasing", allowing the introduction of specific probability curves to demonstrate variance and the Gambler's Fallacy.

The interface is optimized for high-frequency usage with keyboard shortcuts (Spacebar) and instant visual feedback. Accessibility compliance (WCAG 2.1) ensures that screen readers receive immediate state changes, making statistical generation available to all users.

Formulas

The core mechanic of this tool relies on the Bernoulli Distribution for a random variable X, where X=1 represents Heads and X=0 represents Tails.

P(X=k) = p^k(1 − p)^1−k

For Batch Simulations (checking distribution over time), we utilize the Standard Error of the Proportion (SE) to determine if the random number generator is functioning within statistical norms:

SE = √p(1 − p)n

When the "Weighted" mode is active, the acceptance threshold T shifts from the median:

Result =

{

Heads if random() < W_headsTails otherwise

Reference Data

Event Type	Probability Formula (P)	Standard Deviation (σ)	Expected Variance (100 flips)
Single Fair Flip	P(H) = 0.5	0.5	± 5
Sequence (H,H,H)	0.5³ = 0.125	N/A	≈ 12 occurrences
Weighted Flip (70% Heads)	P(H) = 0.7	√p(1−p) ≈ 0.458	± 4.6
At least 1 Head in n flips	1 − (1−p)ⁿ	Variable	Converges to 1
Gambler's Ruin	1 − (q/p)ⁱ1 − (q/p)^N	High Volatility	Dependent on Bankroll
Perfect 50/50 Split	nCk ⋅ p^kq^n-k	Decreases as n ↑	≈ 8% Chance (n=100)
Streak of 10 Heads	(1/2)¹⁰	Rare Event	1 in 1024 trials
Quantum Superposition	1√2(\|H> + \|T>)	Undefined	Observation Collapse

Frequently Asked Questions

Yes. The engine prioritizes `crypto.getRandomValues()` when available in the user's browser. If unavailable, it falls back to a high-entropy implementation of `Math.random()`. This ensures that the outcome is not predictable by state analysis, making it suitable for secure drafting or randomized controlled trials.

Consensus Mode flips 3 or 5 coins simultaneously. This is used to reduce variance in decision-making (best 2 out of 3). It effectively calculates a binomial cumulative probability, ensuring that the "Winner" is determined by the majority state, reducing the likelihood of "fluke" results.

This is a physics-simulation feature utilizing CSS3D transforms and bezier curves. It mimics the conservation of angular momentum where a coin might land on its edge briefly before gravity resolves the vector. It does not alter the pre-calculated outcome but provides visual feedback on the chaotic nature of the toss.

Yes. Enable the "Session Log" in the sidebar. This tracks every flip outcome, the exact timestamp, and the calculated delta. You can copy this data to a spreadsheet to run a Chi-Square Goodness of Fit test to verify that the distribution approaches 50/50 over a large sample size.

If the weight is set to 100% Heads, the `random()` check will always be lower than the threshold (1.0). The system includes a sanity check; if you set a weight to 100%, the animation will still play, but the outcome is deterministically forced. This is often used for "Loaded Dice" demonstrations in probability classes.