> When I lose, I lose more than I win.
I don't see how this is true. Winning adds 0.5x of your current bankroll. Losing subtracts 0.4x of your current bankroll. When you win, you win more than you lose. The gamble has positive expected value.
I don't see how this is true. Winning adds 0.5x of your current bankroll. Losing subtracts 0.4x of your current bankroll. When you win, you win more than you lose. The gamble has positive expected value.
You need to gain 67% ($400 on $600 or $667 on $1000) to break even before or after a 40% loss. The math is 1/(1-0.4)-1=0.67 or (1-0.4)*(1+0.67)=1.
> But this is purely a result of the distribution of returns from a single toss.
Your examples contain two tosses.
Your stake should grow or shrink by the same factor if you win or lose.
If you win, your stake is multiplied by 1.5.
If you lose, it should then be divided by 1.5 (multiplied by 0.66..).
But is in fact multiplied by only 0.6.
> If you lose, it should then be divided by 1.5 (multiplied by 0.66..).
In what sense "should" this be true?
Of course the actual amounts, percentages, proportions look bigger for the win compared to the loss, but that's just maths.
HH: 2.25! HT: 0.9 TH: 0.9 TT: 0.36.
Expectation: 4.41/4 = 1.1025. The losses due to HT/HT get more than offset by the massive gain from HH over TT.
I can imagine there are arguments to be made about median expected value, and the effect on concentration of wealth. But whatever they are, they aren't being made.
Put another way, for each HH pair you should expect a TT pair, and a HHxTT or TTxHH sequence is worth the same as HTxHT, losing the usual 10% every two tosses. 2.250.36=0.81=0.90.9.
The thing that still confuses me is, why the heck is the EV 1.05? It seems to be expressing something true - if you were to split your money into a thousand piles and "play" each individually, you make money overall.
The article, just like the poster above you, characterizes a series of bets into win/loss pairs that add up to a 0.9 return per pair. There are lots of sequences that can be characterized this way. However there will be a few sequences that contain many, many heads and win a lot of money. There are of course also a few sequences with many tails, but their loss cannot decrease below zero so it is contained.
So it's a little bit like a lottery ticket, where the positive gains are extremely concentrated into a very small lucky group. The more rounds of the gamble you play, the smaller the lucky winning group gets, and the larger their wealth.
This is exactly the catch of the experiment. It's proposing you cannot split your money in independent experiments. You need to pick a history and stick with it. That's what ergodicity is about.
Hence, no one owns the average (EV) money of all possible outcomes and it's a pointless metric, even though it doesn't say something that's mathematically false.
2.25 * 0.36 = 0.81 = 0.9 * 0.9.
HH: 2.25 HT: 0.9 TH: 0.9 TT: 0.36.
Expectation: (2.25 * 0.9 * 0.9 * 0.36)^(1/4) = 0.9. So you expect to lose 10%. The losses don't get offset by the wins. Intuition: Possibility of losing 100% is not offset by a possibility of winning 100%, so the arithmetic mean is wrong.