s ▁are ▁always ▁the ▁same . ▁ ▁In ▁the ▁he ur istic ▁proof ▁above , ▁ ▁success es ▁and ▁ ▁fail ures ▁are ▁highly ▁likely ▁only ▁for ▁very ▁large ▁. ▁ ▁Using ▁Python ▁and ▁Sym Py ▁ ▁For ▁a ▁symbol ic ▁ver ification ▁with ▁Python ▁and ▁Sym Py ▁one ▁would ▁set ▁the ▁derivative ▁ ▁of ▁the ▁expected ▁value ▁of ▁the ▁log arith mic ▁bank roll ▁ ▁to ▁ 0 ▁and ▁solve ▁for ▁: ▁ ▁>>> ▁from ▁sym py ▁import ▁* ▁>>> ▁x , ▁b , ▁p ▁= ▁symbols (' x ▁b ▁p ') ▁>>> ▁y ▁= ▁p ▁* ▁log ( 1 ▁+ ▁b ▁* ▁x ) ▁+ ▁( 1 ▁- ▁p ) ▁* ▁log ( 1 ▁- ▁x ) ▁>>> ▁solve ( diff ( y , ▁x ), ▁x ) ▁[ -( 1 ▁- ▁p ▁- ▁b ▁* ▁p ) ▁/ ▁b ] ▁ ▁Bern oul li ▁▁ ▁In ▁a ▁ 1 7 3 8 ▁article , ▁Daniel ▁Bern oul li ▁suggested ▁that , ▁when ▁one ▁has ▁a ▁choice ▁of ▁bet s ▁or ▁invest ments , ▁one ▁should ▁choose ▁that ▁with ▁the ▁highest ▁geometric ▁mean ▁of ▁out comes . ▁This ▁is ▁mathemat ically ▁equivalent ▁to ▁the ▁Kelly ▁criter ion , ▁although ▁the ▁motiv ation ▁is ▁entirely ▁different ▁( B ern oul li ▁wanted ▁to ▁resolve ▁the ▁St . ▁Petersburg ▁par ado x ). ▁ ▁An ▁English - language ▁translation ▁of ▁the ▁Bern oul li ▁article ▁was ▁not ▁published ▁until ▁ 1 9 5 4 , ▁but ▁the ▁work ▁was ▁well - known ▁among