Is anyone aware of the number of submissions to YC per year and the chance of getting an interview?
It took me a long time to realize that when the odds of getting into something were described as e.g. 1 in 10, that didn't mean the odds for any given applicant were 10%, but rather (to the extent the people deciding were good judges) that for 10% of applicants the chance was nearly 100%, and for the other 90% nearly zero.
The famous example mentioned is W.S. Robinson's 1950 study of literacy among immigrants[1]. He found that states with higher populations of immigrants had higher literacy rates, but the average literacy rate among individual immigrants tended to be less than the average population. He concluded that immigrants must move to areas where the literacy rate was higher.
It's not too much of a stretch to make a similar argument for Silicon Valley as a whole or refute any individual startup's odd's of success as 1 in 20.
A simple graphic illustration of this is smoking. A smoker's probability of dying of a smoking-related disease before age 65 is 15.6%[1]. However, my probability of dying of a smoking-related disease (assuming I smoked) is either 0% (I don't die of a smoking-related disease) or 100% (I die).
People don't understand why some people smoke when there is such clear evidence of the increased probability of death due to smoking-related disease. Well, for each smoker, the probability is either 0% or 100%. If the smoker believes his probability is 0%, he will continue to smoke. If he believes his probability is 100%, he is a hypochondriac.
Thus, smokers either believe they are untouchable or they are crazy.
This is why it is so hard to sell "it's good for you" things... they are almost invariably statistically good for a large population, but can make no guarantees of "goodness" when applied to a singleton.
This applies in spades to health-anything:
* Individual's health: Lose weight and exercise - it's good for you.
* Program's health: testing is not guaranteed to find any bugs - if it were, running testing a second and third time would always find more bugs.
[1] http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1646951/?page=3