An investor wishes to invest $\$700$.
There are two independent stocks the investor can choose to invest in, both of which are currently trading at the same share price. The daily returns of the first stock are historically normally distributed with a mean of $3\%$ and a standard deviation of $1.5\%$. The daily returns of the second stock are historically normally distributed with a mean of $4\%$ and a standard deviation of $2\%$.
How much should the investor choose to invest (in dollars) in the first stock to maximize his probability of having a positive profit over the course of a day?
The answer says when the standard deviations of returns are the same, the probability of profiting is maximized. Suppose he invests \$$x$ into the first stock, then $1.5x = 2(700-x)$, so $x = 400$.
But why does this happen when the standard deviations are equal?