▁expected ▁net ▁w inn ings ▁from ▁a ▁$ 1 ▁bet , ▁since ▁the ▁two ▁out comes ▁are ▁that ▁you ▁either ▁win ▁$ ▁with ▁probability ▁, ▁or ▁lose ▁the ▁$ 1 ▁w ager ed , ▁i . e . ▁win ▁$ − 1 , ▁with ▁probability ▁. ▁Hence : ▁▁▁ ▁For ▁even - m oney ▁bet s ▁( i . e . ▁when ▁), ▁the ▁first ▁formula ▁can ▁be ▁simplified ▁to : ▁▁ ▁Since ▁, ▁this ▁simpl ifies ▁further ▁to ▁▁▁ ▁A ▁more ▁general ▁problem ▁relevant ▁for ▁invest ment ▁dec isions ▁is ▁the ▁following : ▁The ▁probability ▁of ▁success ▁is ▁. ▁If ▁you ▁succeed , ▁the ▁value ▁of ▁your ▁invest ment ▁increases ▁from ▁ ▁to ▁. ▁If ▁you ▁fail ▁( for ▁which ▁the ▁probability ▁is ▁) ▁the ▁value ▁of ▁your ▁invest ment ▁decre ases ▁from ▁ ▁to ▁. ▁( Note ▁that ▁the ▁previous ▁description ▁above ▁assumes ▁that ▁ ▁is ▁ 1 .) ▁ ▁In ▁this ▁case , ▁as ▁is ▁proved ▁in ▁the ▁next ▁section , ▁the ▁Kelly ▁criter ion ▁turns ▁out ▁to ▁be ▁the ▁relatively ▁simple ▁expression ▁▁▁▁ ▁Note ▁that ▁this ▁reduces ▁to ▁the ▁original ▁expression ▁for ▁the ▁special ▁case ▁above ▁() ▁for ▁. ▁ ▁Clear ly , ▁in ▁order ▁to ▁decide ▁in ▁favor ▁of ▁invest ing ▁at ▁least ▁a ▁small ▁amount ▁, ▁you ▁must ▁have ▁▁▁▁ ▁which ▁obviously ▁is ▁nothing ▁more ▁than ▁the ▁fact ▁that ▁the ▁expected ▁profit ▁must ▁exceed ▁the ▁expected ▁loss ▁for ▁the ▁invest ment ▁to ▁make ▁any ▁sense . ▁ ▁The ▁general ▁result ▁clar ifies ▁why ▁lever aging ▁( t