03 July 2008

Lightning and lotteries

From a rerun of Friends:
Ross: Do you know what your odds are of winning the lottery? You have a better chance of being struck by lightning 42 times.
Chandler: Yes, but there's six of us, so we'd only have to get struck by lightning 7 times.
Joey: I like those odds!
Unsurprisingly, Chandler seems to know that probability doesn't work this way; Joey doesn't.

Also, Ross is wrong. It seems the record for getting struck by lightning is Roy Sullivan, seven times. So nobody's been hit 42 times, while plenty of people have won the lottery.

I don't know how to calculate the odds that someone gets hit 42 times by lightning in their life; the lifetime incidence of getting hit is three thousand to one, and if you figure that lightning strikes are a Poisson process with rate 1/3000 per lifetime, as this article states, then the probability that lightning hits one person seven times is something like one in (1/3000)7/7!, or one in about 1028. (That's the probability that a Poisson with parameter 1/3000 takes the value exactly 7; I'm ignoring the normalizing factor of exp(1/3000) and the even-more-negligible probability that someone gets hit eight or more times.)

Since the number of people who have existed is much less than 1028, the existence of a person who's been hit seven times is very strong evidence that that's not the right model. My hunch is that events of each person getting hit by lightning are a Poisson process, but with a separate parameter depends on the person. Roy Sullivan was a park ranger.

But the 1 in 3000 figure can't be trusted; the article also claims the annual risk of getting hit by lightning is one in 700,000. People don't live 700,000/3,000 (i. e. 233) years.


Anonymous said...

Presumably there is a significant chance of a lighting strike being fatal, so at least for the first 6 strikes, you should be an even smaller probability that is scaled by the survival rate.

Also, for most people, is the constant Poisson parameter assumption valid? I mean if I had been struck, I would definitely take steps to decrease my parameter in the future.

Anonymous said...

Isabel, so I was thinking about the stock market today, and what the expected return for the market is. As you know, thanks to William sharpe, all financial theory is based on the expected return of equities.... except no one can agree on what to "expect". Most of the time its the arithmetic mean of past returns, but sometimes people use the geometric mean, still others the arithmetic mean plus dividend yield. No one seems to use the harmonic mean, that i've encountered.

So my question is more general. What can the difference between the arithmetic mean and the geometric mean (and harmonic) tell us about the dispersion of the data we are looking at?

also, the harmonic mean seems to relate to the exponential function,as does the normal distribution, which brings me back to my question about what difference between the means tells us about dispersion.


I. J. Kennedy said...

Krusty: You people are pigs! I, personally, am going to spit in every 50th burger!

Homer: I like those odds.