Combats in the Republic of Rome Game are resolved using the results of the sum of rolling three dice, otherwise known as a TDR. The minimum roll of three dice is a "3" and the maximum is "18".  If one thinks of each die as a separate entity, say three differently colored dice are being rolled, then there are 216 possible die roll combinations that lead to the resulting sums ranging from "3" to "18" inclusive. This first chart shows what chance there is in 216 combinations of reaching the given sums on a TDR.  It also includes the percentages attained by adding all sums from "3" to a given higher sum (% increasing) and the reverse of that (% diminishing) for a given sum to number "18".  For example, the odds of rolling sums of any number from 3 through 6 =  9.26%, whereas the odds of rolling any sums from 8 through 18 = 83.80%.  Note the symmetry in the second and third columns, which show that whereas the end sums of "3" or "18" have the least chance of being rolled, those sums totalling 10 or 11 have the greatest chance of being rolled (i.e., a 25% chance that "10" or "11" will be rolled).
 

The following chart applies the percentages in the last two columns of TDR Chart #1 to demonstrate the variable chance of defeat or victory as the combat modifier increases or decreases.  These are applied to the full range of defeat or victory results, and also to the range of specific results for Defeat-All Units lost, and for Victory-No Losses.  Stalemates, since they include varying ranges in the center of the chart without necessarily including the end sums of "3" or "18" had to be calculated separately.

TDR Chart #2 deals specifically with the combat modifiers which are added to the TDR during combat in the Republic of Rome game. The "0" row represents the unmodified results as depicted exactly on the game's Combat Result Table.  As the Romans outnumber their opponents in overall strength by the indicated modifiers, Roman chances of victory increase.  Chances of defeat increase when the enemy has greater strength than Rome.  The portion of the chart which includes negative modifiers will tend to be used when a weak province must attack a strong war, or when a proconsul with diminished forces must continue to fight a war.  The chart clearly shows that when the combat modifier is +11 or more, the only possible result is a victory, although the modifier must be +15 (or greater) to guarantee victory with no losses.  The third useful modifier is +5, since a modifier of +5 (or more) guarantees that there will be no defeat.

The chart shows the exact range of die roll sums which must be rolled to produce the indicated percentage results.  Note that the percentages for Defeat + Stalemate + Victory always should total to 100% (or at least 99.99%).  Chart #2 is thus useful as a general guide to chances of defeat, stalemate, and victory, normally with the caveat "barring Disaster/Standoff results."  Nevertheless, some statesmen (e.g. Scipio Africanus, Quintus Flamininus, and Aemilius Macedonicus, etc.) have a combat ability which allows Disaster/Standoff results to be voided against specific wars.  In such a case, the following chart needs no modification.
 

In Chart #3, the emphasis is on the upper part of Chart #2 in which the Romans have a chance at Victory. The TDR sums are read across the chart, and the combat modifiers run down the left side.  Some of the abbreviations for the combat results as shown in the chart have been made bold for emphasis so that the chart is easier to read.

Key: Vn = Victory, no losses    V = Victory    S = Stalemate    D = Defeat    Da = Defeat, all units lost
 

In Chart #4, the emphasis is on the lower part of Chart #2 in which the Romans have a chance at Defeat. The TDR sums are read across the chart, and the combat modifiers run down the left side.  Some of the abbreviations for the combat results as shown in the chart have been made bold for emphasis so that the chart is easier to read.

Key: Vn = Victory, no losses    V = Victory    S = Stalemate    D = Defeat    Da = Defeat, all units lost
 

My thanks to colleagues Carolyn Eberly and Beth Petsche for their assistance with the initial calculations for Chart #1. Any errors should be directed to the attention of the designer of these charts:  Lou Jerkich.