On Mon, 2004-10-04 at 10:42, Stathis Papaioannou wrote: > Eric Cavalcanti writes: > > QUOTE- > And this is the case where this problem is most paradoxical. > We are very likely to have one of the lanes more crowded than > the other; most of the drivers reasoning would thus, by chance, > be in the more crowded lane, such that they would benefit from > changing lanes; even though, NO PARTICULAR DRIVER would benefit > from changing lanes, on average. No particular driver has basis > for infering in which lane he is. In this case you cannot reason > as a random sample from the population. > -ENDQUOTE > > I find this paradox a little disturbing, on further reflection. You enter > the traffic by tossing a coin, so you are no more likely to end up in one > lane than the other, and you would not, on average, benefit from changing > lanes. Given that you are in every respect a typical driver, what applies to > you should apply to everyone else as well. This SHOULD be equivalent to > saying that if every driver decided to change lanes, on average no > particular driver would benefit - as Eric states. However, this is not so: > the majority of drivers WOULD benefit from changing. (The fact that nobody > would benefit if everyone changed does not resolve the paradox. We can > restrict the problem to the case where each driver individually changes, and > the paradox remains.) It seems that this problem is an assault on the > foundations of probability and statistics, and I would really like to see it > resolved.
I found the answer of why you should be more likely to enter in the crowded lane in this case. The answer came after I tried to think about an example for few people (which turned out not to work as I thought it would) Suppose a coin is toss for N people, which enter one of two rooms according to the result. Suppose first N=3. Then it is more likely that I will be in the crowded room, even though there was no particular bias in each coin toss. But still, if I am given the option to change, and if I am in the crowded room, I'll probably still be in the crowded room after I change! Now as N grows large, it is still more likely that I will be in the crowded room, only it is less so. I was neglecting the effect that you make yourself when you enter the room/lane. When N is large and even, it is equally likely that the lane I enter is slower/faster. But it may be that both lanes have same numbers, so my entering will make that lane be the slower, and that's where the effect comes from. If it is odd, and I enter the fast lane, it is possible that they become equal. If I enter the slower lane, it will become even slower. A minute of thought shows that my changing lanes does not affect the result, though, as much as changing rooms does not make me more likely to be in the less crowded when N=3. Therefore it is not a good advice for people to change lanes in this case, even though it is more likely that they are in the slower lane! Eric.
