This chapter is intended to introduce or refresh some useful math concepts for designers.
It is not a math course, math specialists should not read this section, or at their risk !
As a start point, let's play dice.
If we play with only one die, we can get values from 1 to 6 with the same probability (1/6=0.16667). Average value over a large number of runs is 3.5. Probability of getting a value between 3 and 4 inclusively is 2/6. The sum of probabilities is 1, obviously: The die has to give a value, and that value has to be between 1 and 6!
Now if we play with two dice and sum up the values, result can range from 2 to 12. But what about the probabilities? You can write in a table the possible values for one die and for each of these the possible values of the other die. This is easy to check in a spreadsheet like Excel or LibreOffice.org Calc.
You get 36 possibilities. In the rightmost column that sums the dice values, you can see that:
Let's plot probability versus value (just divide above values by 36 so that integral is 1):
Average value per die is still 3.5. But now, probability of getting a value between 3 and 4 (between 6 and 8 for the sum) inclusively is 16/36.
It has been multiplied by 4/3 with respect to the single die play
With three dice, values range from 3 to 18. Again this can be done in a spreadsheet.
There are 6^3 =216 possibilities with the following occurrences:
Average per die is still 3.5 and now probability of getting a value between 3 and 4 inclusively is 104/ 216.
It has been multiplied by 13/9 with respect to the one die play.
Now let's compare the probabilities for 1 to 4 dice:
Doesn't the 4 Dice curve remind you anything ?