I'm attempting to implement a numerical approximation to assess the frequency of outcomes given $N$ independent random variables. There is a slight twist however, each $N$ has a weight,

Each variable has the form:

$f_n=\{probability, weight\}$

I'm having difficulty confirming what the result should be. Here's an example.

$f_1=\{0.6,7\}$

$f_2=\{0.5, 4\}$

$f_3=\{0.4, 5\}$

My objective is to calculate the probability of each permutation's sum of weights; so given 3 functions, there are 8 permutations:

Sum of Weights: Likelihood
0: 9.66%
4: 14.26%
5: 6.47%
7: 14.44%
9: 9.59%
11: 21.44%
12: 9.71%
16: 14.43%

Are these outcomes correct? Maybe there is a closed solution that doesn't require monte carlo methods?

I'm attempting to implement a numerical approximation to assess the frequency of outcomes given $N$ independent random variables. There is a slight twist however, each $N$ has a weight,

Each variable has the form:

$f_n=\{probability, weight\}$

I'm having difficulty confirming what the result should be. Here's an example.

$f_1=\{0.6,7\}$

$f_2=\{0.5, 4\}$

$f_3=\{0.4, 5\}$

My objective is to calculate the probability of each permutation's sum of weights; so given 3 functions, there are 8 permutations:

Sum of Weights: Likelihood
0: 12.03%
4: 11.98%
5: 8.02%
7: 17.98%
9: 8.04%
11: 18%
12: 12.02%
16: 11.93%

Are these outcomes correct? Maybe there is a closed solution that doesn't require monte carlo methods?