The problem with doing 20 small bets vs. one large one is your almost guaranteed to lose on a given day. You are much better off making 1 bet per year and then stopping, than lot's of little ones. EX: I am up over my lifetime on slot machines. How? I did a few bets on a fixed amount got slightly lucky and never played again.
That said, if your doing this for entertainment then minimizing the money your betting is reasonable. People making 100+k per year can get really competitive over penny poker.
(Odds of winning * 20 bets * $1) has exactly the same expected outcome as (odds of winning * 1 bet * $20), assuming the odds are the same in the two brackets, but the difference is in the latter category you get to play 20 times.
The list of possible outcomes changes, but the expected outcome does not.
2x winnnings * 50% chance of winning * 20 games * $1 per game = expected outcome $20
2x winnings * 50% chance of winning * 1 game * $20 per game = expected outcome $20
Your fuzzy logic around "winning big" "losing big" etc doesn't change the basic math of the expected outcomes, which is simply (chance of a thing happening) * (result if it happens).
These are not even games; the expected value is < 1. That said if you like playing then the actual math is almost meaningless. Nobody complains about not breaking even when going to the movies.
With one game you never actually end up with ~20$. It's either 0 or ~40$. Expected value is close to break even, but you get an actual bet going.
With 20 games you never end up at 0 or ~40$ it's 'always' (524,287/524,288) something between those extremes. However, with negative expected outcome games the average case is bad, so you odds of winning are less than 50/50 and your average win is tiny.
Further, many people play not just a fixed number of games, but keep playing until there stake is gone, which tends very strongly toward zero.
Net result, if you play just one game you have a reasonable chance of winning. If you play lot's of games you most likely will keep losing money until you stop.
PS: This is why most casinos don't take big bets; they make far more money from people making lots of small bets over time.
If you assume that the odds of winning are the same, the payoff ratio is the same, and that you don't play again with anything you win, then the expected results are exactly the same.
If someone is going to keep playing with their winnings, they're likely to do so either way.
EV is not the only number, your describing an average and the standard diviation is important.
Ex: if you have a slight negative bias say 50:50 to win 1.999x and play a trillion games your going to flat out lose money with Effectivly zero % chance of winning.
But if you play 1 game with a trillion times the bet, you have a 50% chance of walking away with 1.999x your bet, and a 50% chance of walking away with 0. So about a 50% chance to nearly double your bet, and a 50% chance to lose. Whereas with a trillion smaller games you'll probably end up at about 0.999x your starting money, which is the same on average.
Not sure I understand the math behind your logic - unless you're assuming that you spend an entire year researching that 1 bet to have the best odds. In DFS, that would be like spending the entire off season researching Week 1 stats, so that by game day, you play 1 giant tournament hoping for a big payout, and then not playing the rest of the year.
That still seems to be dangerous because you have no diversification, in lineups or allocation of funds. Would be curious to understand your reasoning more.
Edit: Diversification works with investment when the average return is positive.
These games are negative sum for most players as the house takes part of the pool. For simplicity, let's say it's a 50:50 coin, but you win 1.99 of your stake. With one bet the house takes some of your winnings, but your odds don't change. With 2 bets, if you win both you win, if you lose both you lose, but if you win one (1.99) and lose one (0) you still lose (1.99 + 0) / 2 < 1.
Sure, your odds of losing 'big' went from 50% to 25%, but your odds of winning big went from 50% to 25% and your odds of losing slightly are now 50%. Same thing applies if it's a 10% chance to win 9.99x your state, with 10 bets and 1 when you still lose. However, your odds of a 9.99x win are much smaller. (1 in 10,000,000,000 vs. 1:10)
Granted, if your vastly better than other players you can get a small advantage. But, that's not going to be very significant over a small number of games.
That said, if your doing this for entertainment then minimizing the money your betting is reasonable. People making 100+k per year can get really competitive over penny poker.