Re: Correct answer
Yes, $23,667 is the correct answer assuming that your every bet wins what's expected (impossible situation) and the expected profit is always added to the bankroll before making the next bet.
It's a bit difficult for me to explain this since english isn't my main language. But I'm sure me and whipsaw got the right answer.
This is how it goes:
1st bet
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Bet size: $10,000 * 5% = $500
Expected return: $500 * 1.909 * 0.56 = $534.52
Expected net profit: $34.52
(You EXPECT to win this bet 56 times out of 100, but with particular bet you can obviously lose the whole $500 or net $454.5 profit. What matters is that on average you net $34.52 with every bet)
Expected growth of the bankroll: $10,034.52 / $10,000.00 = 0,3452%
2nd bet
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Bet size: $10,034.52 * 5% = $501.73
Expected return: $501.73 * 1.909 * 0.56 = $536.37
Expected net profit: $34.64
Expected growth of the bankroll: $10,069.16 / $10,034.52 = 0.3452%
As you can see, on average your bankroll grows 0,3452% with every bet. Now it's easy to calculate how big the bankroll will be after 250 bets:
1.003452^250 = 2.366741
With your $10,000 bank, this will be $23,667.
Hope this clears things a bit.
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