Probability models for blackjack poker

For simplicity in calculation, previous analyses of blackjack poker have employed models which employ sampling with replacement. In order to assess what degree of error this may induce, the purpose here is to calculate results for a typical hand where sampling without replacement is employed. It is seen that significant error can result when long runs are required to complete the hand. The hand examined is itself of particular interest, as regards both its outstanding expectations of high yield and certain implications for pair splitting of two nines against the dealer’s seven. Theoretical and experimental methods are used in order to determine whether the calculation can be truncated after the dealer’s fifth card without significant loss of accuracy. Less than one tenth of one percent difference is observed between the fifth card truncated expectation and the experimentally obtained expectation.