Roll customizable dice sets and visualize outcome distributions. Compare theoretical vs empirical probabilities with bar charts.
Configure a set of dice, roll it repeatedly, and compare the observed outcome bars with the corresponding theoretical distribution. The theoretical view describes the long-run probabilities for the selected dice, while the empirical view records what happened in a finite simulation. The common surprise is that possible totals are not generally equally likely. For two six-sided dice, a total of 7 has six ordered face combinations, but a total of 2 has only one.
A concrete input and expected output from the current implementation.
Input
2 six-sided dice; 360 simulated rolls
Expected output
For totals 2 through 12, the theoretical probabilities are 1/36, 2/36, 3/36, 4/36, 5/36, 6/36, 5/36, 4/36, 3/36, 2/36, and 1/36. The theoretical bar for 7 is therefore 16.67%, while those for 2 and 12 are each 2.78%. The empirical bars contain 360 observations, but their individual counts vary from run to run.
Two six-sided dice have 36 equally likely ordered face pairs. Six pairs total 7, while only (1,1) totals 2 and only (6,6) totals 12.
For standard fair dice, the theoretical chart accounts for every possible face combination in the configured set. When outcomes are totals, the probability of a total is the number of ordered combinations producing it divided by the product of the dice side counts. This is why central totals usually have taller bars than extreme totals.
Each simulated trial produces one result from the selected dice set and adds it to the matching outcome bar. Dividing a bar's count by the total number of trials gives its empirical frequency. Increasing the trial count often makes the empirical shape resemble the theoretical shape more closely, but it does not force an exact match.
The simulation is a finite random sample, whereas the theoretical bars describe exact long-run probabilities. Even a correct simulation can contain noticeably high or low outcome counts. More trials usually reduce the relative size of these fluctuations, but no finite trial count guarantees that every empirical bar will equal its theoretical value.
No. For an unchanged dice set, the theoretical probability of each total remains the same. Adding trials only changes the empirical evidence shown by the simulation. It generally produces more stable relative frequencies, although later rolls can still move an individual empirical bar either toward or away from its theoretical percentage.
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