Tuesday, November 20, 2007

The Wrong Probability

I saw a small sign put up by students in the stairwell today. I'd seen it before but ignored it. Today it caught my attention. It said simply:
39% of all Traffic Fatalities are alcohol related.
What is wrong with this probability? It does little to suggest that you should not drive while drunk. The fraction of alcohol-related fatalities might rise simply because overall accidents decline! What fraction of traffic fatalities should be alcohol related, anyway? Aren't traffic fatalities bad, regardless of the cause? We don't really care about the probability that an accident involves alcohol, given that the accident has occurred. What probability would be useful? We would need to know the probability that a driver has a traffic accident, given that he or she has been drinking. Even better would be a probability based on the amount of alcohol consumed. If the sign had said "39% of drivers who have had one more drinks prior to driving have a traffic accident," that would be a very sobering statistic.

Another example of this wrong probability is sometimes encountered when discussing hard drugs. I recall hearing numbers in high school such as "80% of cocaine users used marijuana prior to using cocaine" (I'm making the number up; I'm not sure what the actual cited number was, and it doesn't really matter for making my point). This number was intended to suggest that marijuana is a gateway drug--that if you use marijuana, you will likely go on to use cocaine. The problem is that it's the wrong probability. What we really need to know is the fraction of marijuana users who go on to use cocaine.