Golden Penney Ante
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Authors
Dababneh, Issa
McCulloch, Ryan
Elmer, Mark
Issue Date
2018-03-23
Type
Other
Language
en_US
Keywords
Golden ratio , Memory state machine , Penney ante game
Alternative Title
Abstract
This research is joint work with Mark Elmer of SUNY Oswego. In 1969, Walter Penney introduced his "Penney Ante" game, and in 1973 Craswell developed a memory state machine for computing probabilities in this game. We generalized Craswell’s results for arbitrary coin probabilities. In the analysis of the HHH vs HTH game, we noticed that the Golden Ratio played a role. Subsequently, we defined D(n,k) integers which are analogous to the binomial coefficients. Pascal's Triangle displays the binomial coefficients in a triangle so that many interesting and important identities which relate these integers can be discovered. Our goal was to discover and to prove similar identities in the D(n,k) integers. We display the D(n,k) integers in a triangle similar to Pascal's triangle. Our preliminary analysis showed the existence of many analogous identities. We proved these identities through the techniques of mathematical induction and generating functions.