I am working through the "Math for Quantitative Finance" course on brilliant.org. The following question was given as an example:
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 solution given is:
Suppose he invests $x$ dollars into the first stock and $y$ dollars into the second stock. He will have the maximum probability of profiting when the two standard deviations are equal, i.e. $1.5x = 2y$. This, together with $x+y = 700$ gives $x = 400$.
I understand how to solve algebraic systems of equations, so I understand why, given these two equations, the answer is that the investor maximizes their probability of profiting when investing $400 into the first stock.
However, I do not understand why it is the case that maximum probability of profiting occurs when the std. deviations are equal. What is the theory behind this?