Dice expressions probability calculator

Essential part of many tabletop games are random events. They are often introduced to the game in form of dice rolls. This program can help you to understand those dice rolls. You can give it a dice expression and it computes the probabilities and some statistics for you. It can even deal with some form of depedence using names (see examples).

Examples

In all of these examples I assume that dice rolls are independent discrete (integer) random variables.

• 2d6: roll 2 six sided dice and add the results
• (1d4)d6: first, roll a 4 sided die. What you've rolled will determine how many times you'll roll a 6 sided die.
• 1d4 * 2: roll a 4 sided die and multiply the result by 2
• (1 + 2) * 3: 9 (you don't have to use the dice operator)
• (1d4)d(1d2*4): 1d4 will determine the number of dice and 1d2 * 4 will determine the number of dice sides
• variance(2d6): compute the variance of 2d6
• expectation(2d6): compute the expectation of 2d6
• 5d100 < 50: compute the indicator (i.e. random variable with bernoulli distribution) that 5d100 is less than 50
• 3d6 == 6: compute the indicator that 3d6 is exactly eqaul to 6
• 2d100 in [4, 32]: compute the indicator that 2d100 is in the [4, 32] closed interval
• var roll = 1d20; (roll in [10, 19]) * 2d8 + (roll == 20) * 4d8: roll with a 20 sided die. If we get a number between 10 and 19 (including 10 and 19), roll 2d8, if we get a 20, roll 4d8. Otherwise it's 0. Notice the 2 occurances of roll are not independent. They are considered as one roll.

Pitfalls

• independence: The program works with random variables. Each operation on them assumes their independence. This assumption is quite limiting. Consider following expression: 2 * (1d20 == 19) + 3 * (1d20 == 20). Some misguided assumption clould be that this expression is equal to 2 if we roll a 19 and 3 if we roll a 20. This is not the case. Both subexpressions 1d20 are independent - they are completely different rolls. In this example, we can get 0, 2, 3 or 5. You can use variables to resolve this issue: var X = 1d20; 2 * (X == 19) + 3 * (X == 20)
• int vs real: The program can work only with integer random variables (that is, value of a variable can only be an integer). Therefore it is invalid to use operands on random variables in conjunction with real numbers. For example: 1d6 * 2.5 or even 1d6 + 2.0 are invalid.

Reference

Types

The program internally works with 3 basic types: int (a signed integer value), real (a 64-bit floating point number) and rand_var (a discrete integer random variable). Numbers without a decimal part are treated as int. The program will automatically convert between int and real or rand_var in function calls and operators but not vice versa (it can't convert real to int implicitly).

Functions in dice expressions

Current implementation will automatically add functions in the following list. There is a mechanism to add new functions implemented in C++ (see the add_function method in the environment class). Special type any in this list means that you can use it with any type as long as you substitute the same type for all occurances of any.

• real expectation(rand_var): takes a random variable and computes its expected value
• real variance(rand_var): takes a random variable and computes its variance
• real deviation(rand_var): takes a random variable and computes its standard deviation
• int roll(rand_var): generate a random number from given distribution
• any max(any, any): takes 2 values and computes the maximum (it can be a random variable if any is rand_var)
• any min(any, any): takes 2 values and computes the minimum (it can be a random variable if any is rand_var)
• int quantile(rand_var, real): takes a random varialbe, a probability and computes a quantile (denote X a random varialbe, quantile(X, p) = min{ k : P(X <= k) >= p})

Operators

Operators in this list are sorted by precedence from lowest to highest. All operators are left-associative unless stated otherwise:

1. = (assign operator): non-associative
2. < (less than), <= (less than or equal), != (not equal), == (equal), >= (greater than or equal), > (greater than), in (is in interval): all non-associative
3. + (addition), - (subtraction)
4. * (multiplication), / (division)
5. - (unary minus)
6. d or D (dice roll operator)

Grammar

The program implements a predictive parser for it is easy to write it by hand. Original grammar is listed here. It is necessary to modify the grammar for a straightforward implementation (for example: get rid of the left recursion).

<stmts>        ::= <stmt>; <stmts> 
                 | <stmt>
                 | ""

<stmt>        ::= var <id> = <expr>
                | <expr>

<expr>        ::= <add> in [<add>, <add>] 
                | <add> <rel_op> <add> 
                | <add>
                
<add>         ::= <add> + <mult> 
                | <add> - <mult> 
                | <mult>
                
<mult>        ::= <mult> * <minus> 
                | <mult> / <minus> 
                | <minus>
                
<minus>       ::= -<minus> 
                | <dice_roll>
               
<dice_roll>   ::= <dice_roll> d <factor> 
                | <factor>
                
<factor>      ::= (<expr>) 
                | <number>
                | <func_id>(<opt_params>)
                | <id>
                
<opt_params>  ::= <param_list> 
                | "" 
                
<param_list>  ::= <param_list>, <expr> 
                | <expr> 

Here are some non-trivial terminals and their regular expressions:

• <number>: [0-9]+(\.[0-9]+)?
• <id>: [A-Za-z][A-Za-z0-9_]*
• <func_id>: [A-Za-z][A-Za-z0-9_]*/{whitespace}*\( (that is, same as <id> but there is a ( after it)
• <rel_op>: <|<=|==|!=|>=|>

Project structure

The project is separated into 3 parts: a static library, simple CLI (command line interface) program and tests. The static library is linked with the CLI program and the tests.

Requirements

You will need cmake version 3.0 or later and a C++ compiler that supports C++14 (tested on gcc version 7.2.0).

You'll need doxygen, gcov and lcov in order to build documentation and code coverage report respectively. They are not needed in order to build and run the CLI program and tests.

Build

1. Create a build directory (say build) in the project root
2. In this directory run cmake .. (use the -G option to specify generator)
3. Compile (for example: run make if you've used Unix Makefiles)