In fact, to calculate the growth of a bankroll over a time it is not correct to simply raise 1 plus edge times bet size to the number of trials power. This should be clear when you consider the results obtained using this method when hitting 50% over 2 trials given payout odds of +99,900 (decimal: 1,000.0), and a 1% of bankroll bet, implying an expected return of 499% of bankroll per bet (obviously completely unrealistic, but I'm just trying to demonstrate a point).sean1 said:If you start with 10,000$ and bet 1% per game and hit 60% at -110 and make 1000 bets/year, guess what you have in 10 years? $87,600,000,000 That's 87 billion.
i come up with $20.5 billion, but your point is correct.
1.0014545 raised to the 10,000th power on a $10k bankroll = $20.5b
By the methodology given above the resulting bankroll would be $10,000*(1+499%)<sup>2</sup> = $358,801.
However, the actual resulting bankroll would only be $108,801.<sup>*</sup>
In general after n bets, np of which were winners, a starting bankroll of B<sub>0</sub>, a wager per trial of X (as percentage of bankroll), and decimal odds of O, the resulting bankroll, B is given by:
B = B<sub>0</sub> * (1 + (O-1)*X)<sup>np</sup> * (1-X)<sup>n(1-p)</sup>
Hence, the actual resulting bankroll starting with $10,000 after hitting 60% across 10,000 bets, at payout odds of -110 (decimal: 1.9091) would be:
B = $10,000 * (1 + (1.9091-1)*1%)<sup>6,000</sup> * (1-1%)<sup>4,000</sup>
≈ $13,259,897,312
I discuss this in my article on Kelly and expected growth.≈ $13,259,897,312
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<sup>*</sup> If you win the 1st bet and lose the 2nd:
Bankroll after winning 1st bet
= $10,000 + 1%*$10,000*999
= $109,900
Bankroll after losing 2nd bet
= $109,900 - 1%*$109,900
= $108,801
If, you were to lose the 1st bet and win the 2nd your results would of course be identical:= $10,000 + 1%*$10,000*999
= $109,900
Bankroll after losing 2nd bet
= $109,900 - 1%*$109,900
= $108,801
Bankroll after losing 1st bet
= $10,000 - 1%*$10,000
= $9,900
Bankroll after losing 2nd bet
= $109,900 + 1%*$9,900*999
= $108,801
= $10,000 - 1%*$10,000
= $9,900
Bankroll after losing 2nd bet
= $109,900 + 1%*$9,900*999
= $108,801