The algorithm:

Cast $n$ dices, $(d_1,$dots,$d_n),$ and output

$D_n=$:$({\displaystyle \underset{i=1}{\overset{n}{}}}di{)}_6$

$$ PROJECT

$[$Essentially$,$ for reaching generalization of "Theory" for

coins to dices$]$

Given $n$ dices with the Probabilities:

$P_{\text{R}\text{O}\text{B}}(d_i=j)=P(i,j)$:$,1in,0j5.$

Task $1$: We need an efficient $($as fast as you can$)$

algorithm to compute

$Q(n,j)=$:$P_{\text{R}\text{O}\text{B}}((d_1+d_2+$dots$+d_n{)}_6=j),0j5$

Task $2$: We want to run this algorithm for various

values of $\frac{n}{n}$

of

Iask $3$: Analyse how fair is $D_n=$:$(d_1+d_2+$dots$+d_n{)}_6$ :

Compute the "Fairness"

$F=$:${\displaystyle \underset{j=0}{\overset{5}{}}}(Q(n,j)-\frac{1}{6}{)}^{2}on$

those runs. We want it to be "small".

Important Hint for the project: two indopendent

dices

$d_{2}012345$

$q(0)q(1)q(2)q(3)q(4)g(5)$

We need to find but $(6,6)$ matrix $M$ sueh that

$((0),r(1),r(2),r(3),r(4),r(5))=(P(0),P(1),P(2),P(3),P(4),P(5))*M$

darr

$(1,6)=(1,6)*(6,6)=(1,l)$

The entries of matriz $M_1$ depend on the probabilities

$(g(0),q(0),g(2),g(3),g(4),g(5))$ project $$ produet

of natrices...