It's less. A distribution of this type (of normal with more probability in the tails) is the t-distribution. Clearly, if more the the probability are in the tails then there's less in the center.
taken from an article by Nassim Taleb here.
THE WORLD QUESTION CENTER 2006 — Page 17
In it, he highlighted this:
I've enjoyed giving math students the following quiz (to be answered intuitively, on the spot). In a Gaussian world, the probability of exceeding one standard deviations is ~16%. What are the odds of exceeding it under a distribution of fatter tails (with same mean and variance)?
So guys, what's the answer?
The answer from the article:
The right answer: lower, not higher — the number of deviations drops, but the few that take place matter more. It was entertaining to see that most of the graduate students get it wrong. Those who are untrained in the calculus of probability have a far better intuition of these matters.
I still don't get it.