I did a course in probability about a year ago, but things have gotten rusty.
Although I normally hate to have to bother other people with my problems, I need to turn somewhere to get some help. I kind of googled this site up.
I have to toss a coin, and keep getting heads every time. For each time that I continue I will gain a better prize, but if I fail, I loose everything. I can stop at any given point and leave with my prize. So it comes down to me having a 0.5^n chance to get the n:th prize.
Here are the prizes for n successes in a row:
Now let's say that I have a pretty large amount of tries available, say 100 tries. What amount of tosses would be the best to aim for, if I want the highest probability to earn the most.
I would appreciate not only a solution, but also an explanation on why. My math knowledge is pre-university. (2nd grade at Swedish gymnasium)
Thanks in advance for any help!