Several PW meetings ago, I introduced a new set of small unit action rules and both Bob Coggins and Dick Sossi participated in the demonstration. Each turn in the game consists of 3 phases, and certain units have only a 33% chance to move each phase. For example, if the Fighting 44th is so designated, then during Phase 1, percentage dice are rolled and if an 01 to 33 appears, the 44th moves out. If not, it gets another shot during Phase 21 and if it doesn't move on Phase 2, it gets a third and final try on Phase 3 for the elusive 01 to 33. If it fails this third try, then it's tough on the Fighting 44th ... it just doesn't move at all during the turn. Dick Sossi commented during the game that these particular units were useless, because every turn, they only had a 33% chance to move, or do anything. No, said 1, every phase they had only a 33% chance to move, but every turn, since they had 3 bites at the apple, the chance they'd move was somewhat greater. "How much greater?" quothe Sossi. In reply, busy as I was with umpiring my new set of rules, I made a hasty calculation and gave him an obviously flukey answer. Both Coggins and Sossi instantly realized my answer was incorrect. Coggins, being a major diplomatist, said nothing, while Sossi, a killer of men, immediately went for the throat. Flustered, I produced another flukey answer by multiplying incorrectly and was soon reduced to a whimpering, huddled, unloved mass of quivering protoplasm by Sossi's incessant barrage. Needless to say, I don't think I convinced Dick Sossi that the probability per turn was anything other than 33%. A couple of weeks later, at the HMGS March convention, I ran several games using the same concept and was genuinely surprised when several experienced gamers voiced the same thought as Sossi... that is, that each turn, because of the repetitive dice throws at 33%, the probability of moving remains 33%. One way to see that the probability per turn is more than just 33 is to look at 2 players, A and B. We give A only one shot per turn to roll a 33 or less, while we give B, as in the rules, 3 shots. It would seem obvious that A does indeed have a simple 33% chance ... he gets one dice throw and he either makes it or he doesn't. But it should be equally as obvious that B must have more then 33%, for if he doesn't make it on the first throw, he's got a backup throw... and if that, too, doesn't come through, he's got yet a third shot at it. Thus logic would have it that because of B's multiple chances, he's got to have a better percentage of success than A, who's only got one shot. If we examine the possibilities of B's moving sometime during the turn, the following may occur: He can come through on his first roll. The probability for this is simply 33%, i.e., 1/3 success Or he can miss his first shot, but make his second. Missing his first try has a 67% (2/3) probability, after which rolling the dice successfully on his second try has a 33% (1/3) chance. Thus, 2/3 miss x 1/3 success. Or, he can miss both first and second tries (each of which is 67% or 2/3) and make the third 33% roll. Thus, 2/3 miss x 2/3 miss x 1/3 success. Adding up all the above probabilities/ we get the chance of B's doing something other than remaining dormant during the turn: [1/3] + [2/3 x 1/3] + [2/3 x 2/3 x 1/3] = 19/27 = 70.4% chance of success Another way to back into this is to say that B can fail to do anything in only one way: he must miss on the first, and must miss on the second and must miss on the third. But this probability is: 2/3 x 2/3 x 2/3= 8/27, or 29.6% chance to do absolutely nothing. Hence if there's 29.6% chance to do nothing, then the chance of his doing something is 100% - 29.6% = 70.4% which checks the first calculations. As I noted before, I was quite taken aback when I realized that there's a basic group of gamers out there - with years of experience amongst them -- who have only the barest concept of probability, how to compute. Back to PW Review May 1985 Table of Contents Back to PW Review List of Issues Back to MagWeb Master Magazine List © Copyright 1985 Wally Simon This article appears in MagWeb (Magazine Web) on the Internet World Wide Web. Other military history articles and gaming articles are available at http://www.magweb.com