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       #Post#: 6--------------------------------------------------
       Mathematics of Poker Chapter Q/A and simulations
       By: xxHaZ Date: May 27, 2014, 12:56 pm
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       Chapter 1
       Probability p(x) = lim (N -> inf) n0/ n
       Independent (joint probability) vs Dependent (conditional
       probability)
       P of A or B / P of A and B
       for mutually exclusive events (OR) : p(A) + p(B)
       for independent events (AND): p(A)p(B)
       for all events (OR): p(A) + p(B) - p(A)p(B) (for mutually
       exclusive events where p(A)p(B) = 0)
       for dependent events (AND): p(A)p(B|A)
       probability distribution: <P> = sigma pi*xi
       xi = value, and pi = probability
       A = {AA, KK, QQ, JJ, AKo, AKs}
       B = {AA, KK, QQ}
       <A, B> : expectation for playing the distribution A vs
       distribution B
       <A, AA|B> : expectation for playing the distribution A against
       the hand AA taken from the distribution B.
       <AA|A, AA|B> : expectation for playing AA from A against AA from
       B
       <A, B> = p(AA)<A, AA|B> + p(KK)<A, KK|B> + p(QQ)<A, QQ|B>
       Key concepts:
       - the probability of an outcome of an event is the ratio of that
       outcome's occurence over an arbitrarily large number of trials
       of that event.
       - a probability distribution is a pairing of a list of complete
       and mutually exclusive outcomes of an event with their
       corresponding probabilities
       - the EV of a valued probability distribution is the sum of the
       probabilities of the outcomes times their probabilities
       - EV is additive
       - if each outcome of a probability distribution is mapped to
       numerical values, the expected value of the distribution is the
       summation of the products of probabilities and outcomes
       - a mathematical approach to poker is concerned primarily with
       the maximization of EV
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