3 ms·
The definition of expectation is what it is :-) I agree that expectation may not be the most meaningful number here, in the long run is almost sure that we los
by kgwgk 7y ago
The definition of expectation is what it is :-)
I agree that expectation may not be the most meaningful number here, in the long run is almost sure that we lose everything but there is a zero probability of infinite gains that makes up for it!
- gowld 7y ago1.05^n is > 1, with positive finite expectation at every finite time step. Why do you say almost sure to lose everything?
- kgwgk 7y agoFor any threshold (say $1, assuming you start with $1mn) and any confidence level (say 99%) there is a number of flips n such that you have over 99% probability of holding less than $1 after playing n times. On the other hand, the expected value of your wealth at that time is 1.05^n million dollars.
- mturmon 7y agoThis is well-stated. In fact, an even stronger statement is possible: If you start with $1MM, eventually you will fall below $1 and never go above $1 again no matter how long you keep playing. And all of that not just on a fraction of sequences, but with probability 1.
- kgwgk 7y agoTrue, as I said in a previous comment we have “almost sure” convergence to zero (i.e. the sequence will converge to zero with probability one).