Quick guide
How to use this calculator
- Select the stated protocol, strategy, approximation, or conversion mode when one is available.
- Enter the labelled finite bound, probabilities, payoffs, sequence state, or special-function argument.
- Read the theoretical or deterministic result separately from any seeded empirical result.
Calculation method
Apply the stated mathematical model
If the first head on toss k≤N pays 2ᵏ and an all-tail cutoff pays 0, the expectation is N.
The calculator validates model hypotheses, finite work limits, probability domains, singular cases, and numerical approximation ranges before returning a result.
Worked example
Worked example
A 20-toss truncation has expected payoff 20 units.
If the first head on toss k≤N pays 2ᵏ and an all-tail cutoff pays 0, the expectation is N.
Supported inputs
Precision and limits
Model scope
Paradoxes depend on their information or randomization protocol. Queueing, games, fractals, and cellular automata use exactly the visible assumptions; changing those assumptions changes the answer.
Deterministic versus simulated
Exact combinatorial and theoretical probabilities are labelled separately from seeded pseudorandom simulations. Reusing a seed reproduces the same finite experiment.
Numerical methods
Finite double-precision arithmetic is used for probability models and special functions. Gamma uses a bounded Lanczos approximation, Bessel J uses a bounded convergent series, and erf/erfc disclose their approximation error.
Limits
Iteration, grid, candidate, trial, and series caps are visible in field labels or calculator-specific notes. A bounded trace never claims to prove an unresolved conjecture.
Calculator-specific rule
The untruncated idealized game has divergent expectation; this page does not prescribe a fair ticket price.