▁that ▁the ▁new ▁sequence ▁ ▁is ▁non - in cre asing . ▁ ▁Thus ▁ ▁will ▁be ▁the ▁best ▁bet . ▁Step ▁ 3 : ▁Set ▁ ▁( the ▁empty ▁set ), ▁, ▁. ▁ ▁Thus ▁the ▁best ▁bet ▁ ▁will ▁be ▁considered ▁first . ▁Step ▁ 4 : ▁Re peat : ▁If ▁ ▁then ▁insert ▁- th ▁outcome ▁into ▁the ▁set : ▁, ▁re calcul ate ▁ ▁according ▁to ▁the ▁formula : ▁▁▁ ▁and ▁then ▁set ▁, ▁Otherwise , ▁set ▁ ▁and ▁stop ▁the ▁repet ition . ▁ ▁If ▁the ▁optimal ▁set ▁ ▁is ▁empty ▁then ▁do ▁not ▁bet ▁at ▁all . ▁If ▁the ▁set ▁ ▁of ▁optimal ▁out comes ▁is ▁not ▁empty , ▁then ▁the ▁optimal ▁fraction ▁ ▁to ▁bet ▁on ▁- th ▁outcome ▁may ▁be ▁calculated ▁from ▁this ▁formula : ▁ ▁. ▁ ▁One ▁may ▁prove ▁that ▁ ▁where ▁the ▁right ▁hand - side ▁is ▁the ▁reserve ▁rate . ▁Therefore ▁the ▁requirement ▁ ▁may ▁be ▁interpreted ▁as ▁follows : ▁- th ▁outcome ▁is ▁included ▁in ▁the ▁set ▁ ▁of ▁optimal ▁out comes ▁if ▁and ▁only ▁if ▁its ▁expected ▁re venue ▁rate ▁is ▁greater ▁than ▁the ▁reserve ▁rate . ▁The ▁formula ▁for ▁the ▁optimal ▁fraction ▁ ▁may ▁be ▁interpreted ▁as ▁the ▁excess ▁of ▁the ▁expected ▁re venue ▁rate ▁of ▁- th ▁horse ▁over ▁the ▁reserve ▁rate ▁divided ▁by ▁the ▁re venue ▁after ▁dedu ction ▁of ▁the ▁track ▁take ▁when ▁- th ▁horse ▁wins ▁or ▁as ▁the ▁excess ▁of ▁the ▁probability ▁of ▁- th ▁horse ▁winning ▁over ▁the ▁reserve ▁rate ▁divided ▁by