Answer to Question #41953 in Statistics and Probability for ALOHA

Question

Consider a purely probabilistic game that you have the opportunity to play. Each time you
play there are n potential outcomes x1, x2, ..., xn (each of which is a specified gain or loss of
euros). These outcomes x1, x2, ..., xn occur with the probabilities p1, p2, ..., pn respectively
(where p1 + p2 + ... + pn = 1.0 and 0 <= pi <= 1 for each i).

Positive xi values mean a gain of |xi| euros and negative values mean a loss of |xi| euros.
Assume that x1, x2, ..., xn and p1, p2, ..., pn are all known quantities. 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 M be the game's expected value. That is, M = p1*x1 + p2*x2 + ... + pn*xn. Let S be the
game's standard deviation. That is, S = SquareRoot( p1 * (x1 - M)^2 + p2 * (x2 - M)^2 + ...
+ pn * (xn - M)^2 ).
In the real world, should a rational player always play this game whenever the expected
value M is not negative? Yes/NO
Explain your answer: