Above Average Number of Fingers
There's a common wisecrack that 90% of drivers consider themselves above average. (A recent mention of this appears in NY Times writer David Pogue's blog, April 29, 2010.) This may sound absurd, but maybe not: After all, the vast majority of people have an above average number of fingers.
More than one of the following assertions explain this fact; which ones?
(1): Signs point to no!
(2): The eccentric opinions of pedants don't matter, since they won't affect the way we count fingers.
(3): This statement means that the average number of fingers is less than ten, which is just a restatement of the claim using the obvious fact that the vast majority of people have the usual ten fingers. But restating the claim does not explain why it is true.
(4): Markov's Theorem has no apparent relevance to the claim. Markov's Theorem simply states that the fraction of people who make up the majority, multiplied by the above average number of fingers, namely 10, cannot be greater than the average. So if 99 percent of the world has 10 fingers, Markov's Theorem implies that the average number of fingers is at least 9.9, which says nothing about how the average compares to 10.
(5): & (6): As long as the total number of missing fingers in the population is larger than the total number of extra ones - it it does not matter how much larger - the average number of fingers will be less than ten.