Two blogs ago I wrote about the Lottery Paradox. I showed that it was false. However, it stayed straying through my mind, not because I had my doubts whether it was really false, for it simply is. But I wondered what went wrong with the paradox and why it is still seen as valid by some. Well, I cannot give an answer to the latter, but I can say something about the former. This time I shall be less abstract and formal, so that those readers who got stuck halfway two weeks ago, will now keep hanging on my lips.
The Lottery Paradox says that we can argue that no ticket will win in a lottery, although certainly one ticket will do, if the lottery is fair. What went wrong in this reasoning besides that the statistical argument isn’t correct? I think that the essence of the failure is in the first basic principle. It runs, as you’ll remember: “If it is highly probable that p, then it is rational to believe that p.”
The central concepts in this principle are “probable” and “rational”. But what do these concepts mean? In the argument that is supposed to substantiate the Lottery Paradox they are not explained. I think that this is the real reason that the argument goes wrong. Let’s look first at “probable”. In the context of the paradox it has a double meaning. First it is treated as a psychological concept but next as a concept from the probability theory (or from statistics). The first principle of the Lottery Paradox says something like this: If it is very likely that p will happen, you can suppose that it really will, even though sometimes it doesn’t. For instance: The timetable says that the next train will leave within 15 minutes, and since the timetable is usually correct, I can better go to the station now (even though it may be possible that the rain will be too late this time). But then, in order to “prove” the Lottery Paradox, “probable” gets suddenly a statistical meaning, and then the argument is false, as I explained in my blog two weeks ago. This doesn’t alter the fact, though, that the psychological interpretation makes sense in our daily life.
There is also something wrong with the way the concept of “rational” is used in the “demonstration” of the paradox. What is rational depends largely on the situation where we have to act. Take the train example again. Suppose that I want to do some shopping in a town nearby. So, I think: “I must leave home now, although it might happen that the train doesn’t leave within 15 minutes, because it will be late or because the timetable has changed”. Then it’s rational to go now and not to check possible changes on the Internet in case there is a train every 15 minutes. If I am wrong, the consequences are negligible.
Take now this example from my blog two weeks ago: I work as a security officer on an airport where I check the passengers at the gate. Say every year ten million passengers pass this airport and only once in five years someone is caught who might have the intention to put a bomb in a plane. Therefore, it is highly likely that the next passenger is a decent person and not a terrorist. Must I say then: Well, it is very, very likely that the next person is not a terrorist. I am a rational person and I don’t check her? Of course not, for in view of the consequences in case she is, it is better to check her, and the next passenger, and the next … Here it is rational not to believe that p, even if it is extremely probable that p.Don’t define your concepts and you can get any conclusion you like.