▁the ▁general ▁case ▁proceed s ▁as ▁follows . ▁ ▁In ▁a ▁single ▁trial , ▁if ▁you ▁invest ▁the ▁fraction ▁ ▁of ▁your ▁capital , ▁if ▁your ▁strategy ▁succeed s , ▁your ▁capital ▁at ▁the ▁end ▁of ▁the ▁trial ▁increases ▁by ▁the ▁factor ▁, ▁and , ▁like wise , ▁if ▁the ▁strategy ▁fails , ▁you ▁end ▁up ▁having ▁your ▁capital ▁decre ased ▁by ▁the ▁factor ▁. ▁Thus ▁at ▁the ▁end ▁of ▁ ▁tri als ▁( with ▁ ▁success es ▁and ▁ ▁fail ures ▁), ▁the ▁starting ▁capital ▁of ▁$ 1 ▁yields ▁▁▁▁ ▁Maxim izing ▁, ▁and ▁consequ ently ▁, ▁with ▁respect ▁to ▁ ▁leads ▁to ▁the ▁desired ▁result ▁▁▁▁ ▁Edward ▁O . ▁Thor p ▁provided ▁a ▁more ▁detailed ▁discussion ▁of ▁this ▁formula ▁for ▁the ▁general ▁case . ▁There , ▁it ▁can ▁be ▁seen ▁that ▁the ▁substitution ▁of ▁ ▁for ▁the ▁ratio ▁of ▁the ▁number ▁of ▁" success es " ▁to ▁the ▁number ▁of ▁tri als ▁implies ▁that ▁the ▁number ▁of ▁tri als ▁must ▁be ▁very ▁large , ▁since ▁ ▁is ▁defined ▁as ▁the ▁limit ▁of ▁this ▁ratio ▁as ▁the ▁number ▁of ▁tri als ▁goes ▁to ▁infinity . ▁In ▁brief , ▁bet ting ▁ ▁each ▁time ▁will ▁likely ▁maxim ize ▁the ▁wealth ▁growth ▁rate ▁only ▁in ▁the ▁case ▁where ▁the ▁number ▁of ▁tri als ▁is ▁very ▁large , ▁and ▁ ▁and ▁ ▁are ▁the ▁same ▁for ▁each ▁trial . ▁In ▁practice , ▁this ▁is ▁a ▁matter ▁of ▁playing ▁the ▁same ▁game ▁over ▁and ▁over , ▁where ▁the ▁probability ▁of ▁winning ▁and ▁the ▁pay off ▁odd