The Smith's New Baby

A prelude to set the mood: Name the three most common last names in the USA.

Playing around at the "Book of Odds" web site today and came to a paradoxical epiphany. Common last names are not that common.

What led to the conclusion was today's "odds of the day": The odds a baby will be born outside of a hospital was 1 in 115.6 (based on USA figures in 2004); which seemed pretty rare. Looking down my student list of just over 100 names there might be one who had been born outside of a hospital. (one wonders how many were planned, and how many were born in the taxi on the way to the hospital)

But glancing through the list of things that were approximately equally likely, one of them was having a last name of Smith (about 1 in 116.5). I think before this I would have thought, in general, that it is unusual for a baby to be born outside of a hospital, and not unusual for a baby to be born with the name Smith... turns out, the former is more probable than the latter.

I think this is an easy fallacy, and one my intro-stats kids make often. For example, they all know that in 100 flips of a coin, 50 heads is the most common outcome.. but they tend to let themselves think "most common" means it happens frequently, when in fact it doesn't. The probability of exactly fifty heads turns out to be a little under 8%.

By the way, the three most common surnames in the US according to Wikipedia were Smith, Johnson, and Williams (Smith is only number two in Canada).

On a more scary thought, the odds a person in a hospital will be diagnosed with an infection acquired while in the hospital is more than five times as likely, about 1 in 22.

So what is the probability that a baby born this week is born outside of a hospital and named Smith?

The Baby at top was neither of those.