Hi everyone. Recently came across this question in an upper level class and to be honest I am stuck on it. I am not completely sure if this is the correct forum but here it is. Any guidance is appreciated.

Consider a purely probabilistic game that you have the opportunity to play. Each time you play there are n potential known outcomes x1, x2, ..., xn (each of which is a specified gain or loss of dollars according to whether xi is positive or negative). These outcomes x1, x2, ..., xn occur with the known probabilities p1, p2, ..., pn respectively (where p1 + p2 + ... + pn = 1.0 and 0 <= pi <= 1 for each i). Furthermore, assume that each play of the game takes up one hour of your time, and that only you can play the game (you can't hire someone to play for you).

Let E be the game's expected value and S be the game's standard deviation.

1. In the real world, should a rational player always play this game whenever the
expected value E is not negative? Why or why not?

2. Does the standard deviation S do a good job of capturing how risky this game is?
Why or why not?

3. If you personally had to decide whether or not to play this game, how would
you decide?