Heuristic example of Likelihood calculation

Consider a game of five-card stud poker with a standard deck of 52 cards
There are (52 choose 5) possible hands = 2,598,960
                    p(Royal Flush: AKQJ10 in one suit)         =        4 / 2,598,960 = 1 / 649,739
                    p(Full House: 3-of-a-kind, & 2-of-a-kind)  = 3,744 / 2,598,960 = 1 / 693
                    Ratio of Likelihoods = 93.7648
                    Log10Likelihood        = -1.9871

        Either type of hand is (very) unlikely, But a Full House is much less unlikely
        So, any random five-card hand is ~100x more likely to be a Full House than a Royal Flush
                    This is calculable by ordinary probability or likelihood theories,
                        because the model (composition of a standard deck) is known

Now: consider a game of Fizzbin played against an extraterrestrial
                The ET deals a series of five-card hands,
                        from a deck with unknown numbers of cards, suits, cards / suit, & ranks
                        and declares which of you is the winner after each deal

         Probability theory cannot predict p(winning) because the necessary data are lacking
         Likelihood (Bayesian) theory can estimate L(winning)
                   based on accumulated data on the composition of hands, and which win or lose
                        E.g., after the first game the minimum number of suits & cards / suit is known
                                After N games, a reasonable model of the maximum numbers is possible
                       The model might assume [be conditioned on the expectation]
                                that all suits contain equal numbers of cards, as in Poker

                        Additional data might indicate that there is a less common suit, or rank
                               Analysis of the distribution of events could estimated
                                E.g., a Chi-Square test might reject the hypothesis that all suits are equally common


All material ©2021 by  Steven M. Carr