Quote from: RHavar on October 15, 2016, 09:59:31 PM

I think there's an idea is when you "bust" your bonus, it's over. You can never lose.

I guess you've never taken a betcoin.tm bonus! Wink

But assuming that's the case, that you get to play your bonus without touching your deposit, and the bonus is cancelled as soon as you bust the bonus amount, then the EV of the bonus would depend upon the odds, house edge, and stake sizing. You would want to pick a game and a play style which maximizes the variance (since your EV is the average of just the positive results)

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Suppose we put the 1 BTC on a 0.9% chance of 100x payout, and if we win, we put 9 BTC on another 0.9% chance of 100x payout. Those are both 10% house edge bets, and result in us wagering 10 BTC if we win the first bet.

We have a 0.991 chance of busting the bonus: bonus value = 0 BTC
We have a 0.009 chance of winning the first bet (leaving us with 100 BTC), and then:
We have a 0.991 chance of losing the 2nd bet: bonus value = 100 - 9 = 91 BTC
We have a 0.009 chance of winning the 2nd bet: bonus value = 91 + 900 = 991 BTC

So the expected value of the bonus is 0.009 * (0.991 * 91 + 0.009 * 991) = 0.8919 BTC

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To try to pick a 'bad' strategy, what if we bet 1 BTC with a 90% chance of a 1x payout ten times in a row... Those are also 10% house edge bets.

We have a 0.9 ** 10 = 0.3486784401 probability of winning all ten bets, and ending up with our original 1 BTC. That's an EV of 0.3486 BTC.

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Can we do worse than 0.3486 BTC somehow while only risking 10 BTC? Or better than 0.9 BTC?

Suppose we put the 1 BTC on a 0.09% chance of 1000x payout, and if we win, we put 9 BTC on another 0.09% chance of 1000x payout. Those are both 10% house edge bets, and result in us wagering 10 BTC if we win the first bet.

We have a 0.9991 chance of busting the bonus: bonus value = 0 BTC
We have a 0.0009 chance of winning the first bet (leaving us with 1000 BTC), and then:
We have a 0.9991 chance of losing the 2nd bet: bonus value = 1000 - 9 = 991 BTC
We have a 0.0009 chance of winning the 2nd bet: bonus value = 991 + 9000 = 9991 BTC

So the expected value of the bonus is 0.0009 * (0.9991 * 991 + 0.0009 * 9991) = 0.89919 BTC

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This looks like it approaches an EV of 0.9 BTC as the payout multiplier increases.