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Hi, joker21,
I had the same problem the first time I read this article. I believe the author was trying to say that betting 1/2 Kelly without resizing your bets results in a RoR of 1.83%. I believe that is correct. The sentence in question, however, reads:
"Then the probability of doubling a bank before going bust (RoR) is about 1.8%."
I'm pretty sure he means the RoR before doubling is about 1.8%, which is a helluva lot different than the probability of doubling the bank is 1.8%.
As for the math, the formula he shows is all screwed up. His formula in the text shows: (1-13%)^10=17%. This is wrong. This leaves out the first part of the equation and it uses 13% instead of 1.8%. That might be due to the actual RoR being 1.83% and he left out the 8 and, for some reason, a decimal point. I have no idea what went wrong there... The formula probably should read:
1 - ( 1 - 1.83% ) ^ 10 = 16.86%
If you convert 1.83% to a decimal, it is 0.0183. Subtract 0.0183 from 1 and you get 0.9817. If you raise 0.9817 to the 10th power you get 0.8314. Subtract 0.8314 from 1, you get 0.1686. Convert that to a percentage and you get 16.86%. Round that to 17% and you get his answer.
He raises the RoR to the 10th power because he earlier said the player doubled his bank TWICE a year and plays for 5 years, which equals 10 bank doublings. He's saying that the CRoR in that case is 17%. In other words, the player has a 17% chance of failure within 5 years if he does NOT reinvest any of his winnings.
After 10 years, there will have been 20 doubling periods so the CRoR is 31.1%. In other words, there's a 30% chance the player will go bust in 10 years if he does not reinvest any of his winnings.
Continuing his logic, the player has a greater than 50% chance of going bust if he keeps doing this for 20 years.
Frankly, the article is not easy to follow, but the gist is helpful.
I think he's trying to say that the Cumulative RoR is ultimate death if winnings are NOT reinvested but it can be managed (even with bet resizing) by reinvesting a portion of a doubled bank.
In the fabled "Long Run" ANY RoR will ultimately kill you.
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