In a Hidden-Markov Model where there is fair die

             and a loaded die (with Prob(head)=1/3, Prob(Tail)=2/3)

                 and the outcome was, 10, Heads and , 10, tails

          The most reasonable(Bayesian-Laplace style not ML) estimate

                  for the probability of using the fair die is

                                2863343001597125
                                ----------------
                                4873041614471168

                    that in floating point is, 0.5875884567

                In a Hidden-Markov Model where there is fair die

             and a loaded die (with Prob(head)=1/3, Prob(Tail)=2/3)

                 and the outcome was, 20, Heads and , 20, tails

          The most reasonable(Bayesian-Laplace style not ML) estimate

                  for the probability of using the fair die is

                         327381605443283609260043381137
                         ------------------------------
                         500237630717560914135201349632

                    that in floating point is, 0.6544521750

                In a Hidden-Markov Model where there is fair die

             and a loaded die (with Prob(head)=1/3, Prob(Tail)=2/3)

                 and the outcome was, 30, Heads and , 30, tails

          The most reasonable(Bayesian-Laplace style not ML) estimate

                  for the probability of using the fair die is

                12792251351728996089211427953540765767352372363
                -----------------------------------------------
                18197283968799193634658020753335072511875874816

                    that in floating point is, 0.7029758602

                In a Hidden-Markov Model where there is fair die

             and a loaded die (with Prob(head)=1/3, Prob(Tail)=2/3)

                 and the outcome was, 40, Heads and , 40, tails

          The most reasonable(Bayesian-Laplace style not ML) estimate

                  for the probability of using the fair die is

        197442711694271771810537484166464324867464893948112053755137411
        ---------------------------------------------------------------
        267513351741818722858367094244055284106004524619103095033430016

                    that in floating point is, 0.7380667560

                In a Hidden-Markov Model where there is fair die

             and a loaded die (with Prob(head)=1/3, Prob(Tail)=2/3)

                 and the outcome was, 50, Heads and , 50, tails

          The most reasonable(Bayesian-Laplace style not ML) estimate

                  for the probability of using the fair die is

179971448156000456555254178467389100643966216843242522316883872508528795235

       /
      /
     /

    235556526962676761530958292908591768503738161606911513989651378226904170496

                    that in floating point is, 0.7640265820

                          This took, 29.037, seconds