Exact calculation vs. simulation
Distinguish a combinatorial answer from a Monte Carlo estimate.
When an exact calculation applies
A simple opening condition without effects or mulligans follows the hypergeometric distribution. It sums every way to draw the required number of K targets in n cards from an N-card deck without replacement. The same inputs always give the same answer and have no sampling error.
Both success rate and desired-copy rate in simple mode are exact probabilities.
When simulation is needed
Mulligans, searches, top-N selection, discard-and-draw effects, multiple turns, and resource payments create a rapidly growing state space. Advanced mode executes the configured sequence across many virtual games and estimates the success rate.
The result evaluates the play policy you configured. A player who keeps different cards or orders effects differently may produce a different real outcome.
Run a fair comparison
Use the same trial count and seed for two deck versions. If the difference is smaller than the interval width, increase the sample size. Start with a small run to catch setup errors and reserve larger runs for the final comparison.
Result cards preserve the seed, trial count, and 95% confidence interval, so a shared number keeps its uncertainty and can be reproduced.
A practical decision rule
Use exact mode for a plain “at least m of K targets” question. Use simulation when action order or any conditional choice matters. Do not present an estimated rate as if it had the certainty of an exact result.