Re: Great problem solved - original material
Re: Great problem solved - original material – My new slant….
Dear Bruce and others interested in these types of problems,
In elementary probability problems where the probabilities are constant (Bernoulli trials), the use of the Markoff process is not necessary. Like they say: “in fixed or constant probability problems” – the cards, dice, and wheel, “whatever” has no memory. Therefore in a set of three deals of holdem hands, the odds are fixed and never change. For the problem in question: the answer can be determined by the elementary Bernoulli trial formula of sampling with replacement:
P = comb(n,x) * p^x * (1-p)^(n-x)
Where n = 248, p =(1/221)^3, x = number of successes,
and as you posted in Method 5: Answer = 1 – (1-1/221)^3^248
This model used here can be thought of as: 248 sets of three deals (or 3*248 = 744 deals) – each set taken as independent of any other set. The probability is constant so this is a valid assumption. Look at it this way: set 1 = deal 1 + deal 2 + deal 3 – this is one trial; set 2 = deal 4 + deal 5 + deal 6; …. ; last set 248 = deal 742 + deal 743 + deal 744. Like I am trying to say, “this is valid because the cards have no memory,” and simplifies to the method 5 formula: Answer = 1 – (1-1/221)^3^248.
If Methods(1) are also exact, then they should calculate the same answer as method 5. As you know, the choice of 250 deals: 248 sets of 3 deals is purely arbitrary – also the selection of p(AA) = 1/221 is also arbitrary:for Answer = 1 – (1-1/221)^3^248. A simple example problem with flipping a coin where two consecutive heads appears in say 4 trials could easily verify that the simple Bernoulli trial formula works…. This simple problem can be enumerated on an Excel spreadsheet. I would check the boundary (end conditions for method(s) 1 – maybe they are tricky to handle.
Markoff process are generally use in dynamic programming problems where the odds change with time, and recent and past history can be used to determine future actions.
This is just my opinion and hopefully I don’t have too many mistakes….
Regards & stay well
Carl
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