Help with understanding an example from the book 'Fooled by Randomness'

Here's a thought that seems to give results in agreement with some table values and is close to others (see note at end).

For example in the case of quarterly return: It appears that we should take $\mu_{\text{quarter}}=\frac{\mu_{\text{year}}}{4}=.0375$ (measured in return above the baseline of T-bills); and $\sigma_{\text{quarter}}=\frac{\sigma_{\text{year}}}{\sqrt{4}}=.05$

This comes from looking at a year as a sample of size $n=4$ quarters.

The probability of the quarterly $X$ return being above $0$ (i.e., better than T-bills) is computed based on $X\ge 0$ from the quarterly distribution above. This would be a normal distribution, assuming that the distribution for the annual return is normal.

I don't have $100\%$ confidence in this answer, but it works for quarters and months; however, I get a bit of disagreement with the table values as we take finer time subdivisions. (Perhaps this has to do with the look-up for the normal probabilities, and/or rounding of some of the decimals involved.)

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.

I had the same doubts, but if you use 260 instead of 365 for calculating the probability for 1 day you get the exact number (there are about 260 trading days in a year). Also, you have to use 8 hour per day for the same reason.