TITLE: Help with understanding an example from the book 'Fooled by Randomness'
QUESTION [5 upvotes]: This is an example from the book "Fooled by Randomness":
(...)We know a priori that he is an excellent investor, and that he will be expected to earn a return of 15% in excess of Treasury bills, with a 10% error rate per annum (what we call volatility). It means that out of 100 sample paths, we expect close to 68 of them to fall within a band of plus and minus 10% around the 15% excess return, i.e., between 5% and 25% (to be technical; the bell-shaped normal distribution has 68% of all observations falling between -1 and 1 standard deviations). It also means that 95 sample paths would fall between -5% and 35%.
A 15% return with a 10% volatility (or uncertainty) per annum translates into a 93% probability of success in any given year. But seen at a narrow time scale, this translates into a mere 50.02% probability of success over any given second.
Table 3.1 Probability of success at different scales
Scale      Probability
1 year     93%
1 quarter  77%
1 month    67%
1 day      54%
1 hour     51.3%
1 minute   50.17%
1 second   50.02%

How do I calculate the probability of success at different scales (Table 3.1)?
E.g. where does 77% for a quarter come from?

REPLY [3 votes]: I was also reading this book and spent some time to understand it, using the @paw88789 answer, a more formal approach:
$$
\mu_{month}=\frac{15}{12}=1.25\\
\sigma_{month}=\frac{10}{\sqrt{12}}=2.89\\
$$
To X be greater than zero, you have to find the z-score for this point in the normal distribution:
$$
k=\frac{\mu_{month}}{\sigma_{month}}=0.43\\
k=\text{how many standard deviations for zero return}
$$
Once you are looking for the left side of the curve, consider this as a negative value -0.43 $(-0.43*\sigma_{month}\approx-1.25=z_{score})$
And from z-table (z = -1.25), you retrieve the probability of 0.33360, or (1-0.33360) that gives you 0.6664 very close to the book 67%
For the other time scale, please consider @José David Sánchez recommendation for days (260 trading days instead of 365) and 8-hours/per-day as stated also in the book.