Staying in the traffic again this morning, I decided to find a mathematical motivation behind me always being in the slowest lane.
Consider the following formalization: there are 4 empty lanes and 100000 cars. For each car in turn we choose one of the four lanes uniformly at random, and add the car at the end of that lane.
Now, you are in one of those cars. What is the probability that you end up in the longest late? To remove ambiguity, let's say that if two lanes have the same length, the leftmost one is the longest.
Now consider two solutions:
After all the cars are assigned, let's rearrange the lanes such that the longest lane is the first one. Your car was originally assigned to each of the lanes with equal probability. What was the chance it was assigned to the first lane? Well, clearly it was exactly 25%. Hence the answer is 25%.
The probability your car is in a certain lane after all the cars are assigned is equal to the length of that lane divided by the total number of cars. Almost certainly the longest lane has more than 25% of the cars, hence the answer is strictly larger than 25%.
Which of the two solutions is wrong?