2 ms·
For 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
by kgwgk 7y ago
For 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).