                       Basic Probability

 'Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers' - G.H. Hardy, 1940
Introduction
 Understanding probability is vital to design wargames rules, and it can also be a useful asset for developing tactics. (For example a careful analysis of the odds when picking from an army list may well reveal which is the best value troop type for their points). Wherever possible I have used the correct mathematical term, but my memory of statistics is a little rusty. A probability is defined as: 'A number expressing the likelihood that a specific event will occur, expressed as the ratio of the number of actual occurrences to the number of possible occurrences.' Writing down probabilities These are best expressed as a number between 0 and 1, with 0 being something that will never happen and 1 being something that definitely will. This format allows you to easily combine successive events to arrive at overall probabilities. I prefer a decimal format although fractions can also be used. Probabilities are also often expressed as a percentage and to calculate this you can multiply a number on the 0 to 1 scale by 100, thereby giving a number between 0 and 100. Notation The notation used in this section is: * for multiply, / for divide. Fractions are written as 1/2 for half etc.
One dice roll
 Evaluating the probability of a die roll As there is (at least theoretically) an equal probability of the dice landing on any one face, the chance of a success is the total of all the 'successful' faces divided by the total faces. If you need a 5 or more on a d6 then there are two successful outcomes out of 6 possible ones. This can be expressed on the 0 to 1 scale by dividing 2 by 6 giving 0.33. Discreet Events Every die roll is a separate event in that it is not influenced in any way by the results of any other die roll. The commonly used statistical term for this is 'discreet'. Superstition or 'common sense' may make it seem that if you have just rolled 5 sixes in a row, your luck must be 'running out', however your chances of the next die being a six are the same as ever - one in six.
More than one dice
 Average totals One of the easiest statistical calculations to make, is combining a number of probable results to achieve an average (or 'mean' to be more precise). If 35 archers fire an arrow with a 0.1 probability each of hitting, then the average (or mean) number of hits per turn would be 0.1 * 35 = 3.5 hits. Cumulative Events Say the rules require that for a successful hit you must roll a 5 or more on a d6 followed by a 3 or more on a d6. How does that compare for a example to a single roll of a d6 where you have to get a 6? To evaluate the chances of success where multiple rolls are required, all you have to do is multiply them together. So for this example the first roll has a 2 in 6 chance=0.33.The second roll has a 4 in 6 chance=0.66. Multiply these two together and you get 0.22 which could also be expressed as a 22% chance. If there are more rolls then just keep multiplying them together ad infinitum. As all probabilities are 1 or less the cumulative chance can only get smaller.You may have seen a game at fairs where you have to roll 7 sixes on 7 dice to win a car. The chances of this are 1/6 * 1/6 * 1/6 * 1/6 * 1/6 * 1/6 * 1/6 = 1/279936 or 0.000003. If you pay Ł1 to make each roll then on average you will lose Ł279936 for every car that you win.
Glossary
 Factorial the factorial of a number is 1 * 2 * 3 * ... * number. Eg The factorial of 5 is 1 * 2 * 3 * 4 * 5 = 120. In Excel this is given by the formula FACT(x). Mean the sum of a series of values divided by the number of values. Eg 4 is the mean of 2,3,5 & 6 (2+3+5+6 divided by 4). In the everyday world this is known as the average, but in statistics there are other different types of average such as the mode and the median. Product the result obtained by multiplying two or more numbers together. eg 15 is the product of 3 and 5. Sum the result obtained by adding two or more numbers together. eg 7 is the sum of 3 and 4.   