The actual unfair dice distribution

Of course you may want to know the actual distribution of the two dice.

Here is a function that computes it, for any given sum of dice. For reasons that will become clear later, I'll call it a likelihood:

two_dice_likelihood <- function(
    x,                                     # sum of dice to evaluate on
    die_model = rep( 1/6, 6 )          # dice model as in simulate_two_dice
) {
    dice1 = tibble::tibble( dice1 = 1:6, dice1_probability = die_model )
    dice2 = tibble::tibble( dice2 = 1:6, dice2_probability = die_model )
    both_dice = (
        dice1
        %>% cross_join( dice2 )
        %>% mutate(
            sum_of_dice = dice1 + dice2,
            probability = dice1_probability * dice2_probability
        )
    )
    by_result = (
        both_dice
        %>% group_by( sum_of_dice )
        %>% summarise(
            number_of_ways = n(),
            probability = sum( probability )
        )
    )

    result = (
        tibble::tibble( sum_of_dice = x )
        %>% left_join( by_result, by = "sum_of_dice" )
    )

    return( result )
}

For example here's the result for a fair dice:

two_dice_likelihood( 2:12 )

Question

Plot this for fair or unfair dice - does it match your simulation?

Question

Feel free to compute the distribution summary statistics for this distribution.