Played a game the other night at Mr. Z's house... nice figures, nice crowd, simple rules... in all, not too bad. Mr. Z knows quite a bit of military history, is very knowledgeable about history in general, and can discuss the nitty-gritty of numerous battles and who fought 'em, and who won, and why. But Mr. Z has one failing... he knows just about nothing of statistics and probabilities, and he doesn't know he knows nothing of probability or statistics, and this shows up in the rules he writes for his tabletop games. The game that we played had a sequence of (a) writing orders to units, (b) simultaneous moving of troops, (c ) simultaneous fire, and (d) resolution of hand -to-hand combat. Nothing exciting or unusual here, except for the probabilities that showed up in his combat tables. For example, when a unit checked morale, the player took two 6-sided dice and tossed 'em, looking for a 7 or better. That gives a basic probability of passing of 58 percent. I thought that was rather low, and asked Z about the percentage. He replied something to the effect that 7 was the most frequent number to show up in the toss of two dice, which was why he included it in the passing tosses. But how about the overall effect? Why was the level set so low? No comment. Did Z know that he had instituted such a low passing percentage? It didn't appear so... and the other players took the rules as written, and simply tossed their dice. Another instance of Z's apparent lack of knowledge of the dice throwing game appeared in his firing tables. Light infantry units were the most effective on the field... when they fired, they tossed a 6-sided die for every 6-men firing. The resultant toss was the number of hits inflicted on the enemy. And if less than 6 men fired, you'd toss a 6-sided die for each man, and a result of 1 indicated a hit. How do you analyze this sort of system? One way is to look at the average effectiveness of each man firing. (a) For the first 6 men firing, tossing their 6-sided die, the average die roll is 3.5... hence the kill rate of each of the 6 men can be said to be (3.5)/6, or 58 percent. Thus each man, on the average, knocks off a little more than half an enemy figure. (b) For less than 6 men firing, wherein each tosses his own 6-sided die, looking for a 1, the kill rate per man is 1/6, or 16 percent. Note the drop in the per-man effectiveness... it goes from 0.58 down to 0.16, quite a drastic decrease. Looking at the system wherein regular line infantry fired, each 10 men in the line tossed a 10-sided die, and the toss indicated the number of hits on the target. When less than 10 men fired, each would toss his own 10-sided die, looking for a 1 as a hit. (a) For the first 10 men firing, tossing their 10-sided die, the average die roll is 5.5... hence the kill rate of each of the 10 men can be said to be equal to, on the average, of (5.5)/10, or 55 percent. Thus each man again knocks off a little more than half an enemy figure. (b) For less than 10 men firing, wherein each tosses his own 10-sided die, looking for a 1, the kill rate per man is now 1 /10, or a measly 10 percent. There's a third type of unit... grenadiers. For these men, each 8 men in the line tossed an 8-sided die, and the toss indicated the number of hits on the target. When less than 8 men fired, each would toss his own 8-sided die, looking for a 1 as a hit. (a) For the first 8 men firing, tossing their 8-sided die, the average die roll is 4.5... hence the kill rate of each of the 8 men can be said to be equal to, on the average, of (4.5)/8, or 56 percent. And, again, each man knocks off a little more than half an enemy figure. (b) For less than 8 men firing, wherein each tosses his own 8-sided die, looking for a 1, the kill rate per man is now 1/8, or a measly 12 percent. In all instances... light infantry and line infantry and grenadiers ... there's a huge drop in firing effectiveness when less than the "standard" number of men fired ... and I'm pretty sure it wasn't built in purposely, but that Z plucked the procedures from a hat, without really analyzing the consequences, or, for that matter, without knowing how to analyze the system he had produced. Over the years, I've found that an acceptable, playable, casualty rate is in the order of 33 percent... each time a unit fires, it has a one-third chance to do damage to the target. This holds for unit games, and for single-figure encounters. The one-third chance varies... it'll go down if he target is in cover, and it may go up if the firing is at point blank range. Back to PW Review August 2002 Table of Contents Back to PW Review List of Issues Back to MagWeb Master Magazine List © Copyright 2002 Wally Simon This article appears in MagWeb.com (Magazine Web) on the Internet World Wide Web. Other articles from military history and related magazines are available at http://www.magweb.com