Q:

A simple variance question

Say I am paid 1 million dollars per pip on a six sided dice and I roll the dice one time. Now compare this to a game in which I get 1 dollar per pip on a dice, but I roll the die one million times. It seems intuitive to me that a risk averse person would pick the second game, but don't the variances just add because the die rolls are independent leading to the same variance as in the first game? Why is there a difference in variance?

A:

The standard deviation of the first is $$ \sqrt{\frac{35}{12}} 10^6.$$ The standard deviation of the second is $$ \sqrt{\frac{35}{12}} 10^3.$$ Variances add, not standard deviations, and the variance of a constant multiple of a random variable is the constant multiple squared times the variance of the original random variable.
Your intuition is correct that you are more likely to end up near $3.5$ million in the second case and less likely to get $1$ million or $6$ million.

