I have a question regarding adding a risky bet to a low risk portfolio of (completely independent) bets. I could provide a more specific real life problem but I have the feeling I am missing some underlying principles and would like to learn from a generic example and hopefully apply to other problems. So I hope I am providing enough information. Say I have 20 dollars that i want to bet on 20 independent uncorrelated bets. I want to optimize return but I also value low risk as in low variance. I make a string of 15 low risk bets (say a 60% chance of 4 dollar return, 40% of zero). But I choose a much riskier bet for number 16 (say 20% chance of $25 return and 80% chance of zero). The expected value feels sufficient to warrant the extra risk but I assume my portfolio now has higher variance. If my goal is to minimize the standard deviation of the portfolio am I better off making the final 4 bets my typical low risk bets or doing more of these (something in my head is telling me that if i am going to do one of these high risk bets i should do enough to have a better chance of hitting one). If the answer is that it would be better to finish up with more low risk bets, what about this: what effect would it have (strictly on portfolio SD) if I split Bet 16 into 5 20-cent bets which i feel would move my chances of hitting one of these big returns from 20% to more like 67%). Does that lower my portfolio's SD?
You have framed this decision based on picking the portfolio of bets that give you high return (based on the prospective mean return) and low risk (based on the SD of prospective returns).
Given this, a first step would be to find the portfolios that have the least risk for a given level of return. After all, if one portfolio has the same return as another but has lower risk then, without any other considerations, it is preferable.
Having found this set of portfolios you are likely to observe that the higher the return, of the minimum risk portfolio, the greater the risk.
You are then left with a subjective choice between these minimum risk portfolios.
For example, two minimum risk portfolios, A and B say, might be associated with expected returns of x and 2*x respectively and risk of y and 1.5*y respectively. There's nothing inconsistent with my preference for A and your preference for B (I'm just more risk averse than you).
Beyond this you may be interested in looking at expected utility, the "Kelly criterion", and decision theory.
I have answered my own question by learning how to do monte carlo simulations. So, when adding one or more risky bets to a low risk portfolio, the concept of doing a greater number of risky bets in order to increase the chance of hitting one (thereby adding expected value while minimizing the increase in SD) DOES apply - but there is a right way and wrong way to do it. If in my example you had added normal size bet 16 with higher risk/reward, you would continue driving the portfolio SD up by adding more bets like it with 17-20. However, instead of doing Bet 16 as one normal size bet, if you broke it into 5 smaller bets, you would add the same EV (as a single investment of the same size and risk/reward) but with a much smaller increase in portfolio SD.