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I'm looking at a betting game where I have \$100 and want to double my money by repeatedly betting on a biased coin; it shows heads with probability $p<\frac{1}{2}$ in which case I win even money.

I imagine my best strategy is to go all-in, but I also wanted to investigate what happens if I bet a constant fraction $0 of my wealth each toss, so I implemented this on a computer. As I'd never go broke doing this, I actually implemented a bet size of $\max(fw,0.0001)$, where $w$ is current wealth instead. My findings are here (I realise as a new user I'm not allowed to post images):

My question is why does a dip in success probability occur? Shouldn't such a graph be monotone increasing? (I'm hoping this isn't a simple implementation error!)

Any input greatly appreciated, John

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    How many simulations did you carry out to produce the graph? If the number of simulations is too small the dip may be due to the sample of bets being unrepresentative.2012-11-27
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    I did this for 10000 simulations, and I re-did the large f with a smaller minimum bet size than 0.0001; the same results appear to hold.2012-11-27
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    Edward O. Thorp, [The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market](https://www.researchgate.net/publication/247922818_The_Kelly_Criterion_in_Blackjack_Sports_Betting_and_the_Stock_Market), The 10th International Conference on Gambling and Risk Taking, Montreal, June 1997.2016-12-11

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