Probability problem: cars on the road I heard this problem, so I might be missing pieces. Imagine there are two cities separated by a very long road. The road has only one lane, so cars cannot overtake each other. $N$ cars are released from one of the cities, the cars travel at constant speeds $V$ chosen at random and independently from a probability distribution $P(V)$. What is the expected number of groups of cars arriving simultaneously at the other city? 
P.S.: Supposedly, this was a Princeton physics qualifier problem, if that makes a difference. 

As Hagen von Eitzen has pointed out, the number of groups is the number of record low speeds as we scan the cars from the first car to the last. We calculate the expected number in a much simpler way. Label the cars, in order they start out, $1,2,\dots,n$.
For $i=1$ to $n$, let $X_1=1$ if car $i$ is slower than all the cars $j$ with $j\lt i$, and let $X_i=0$ otherwise. Then the number $Y$ of record low speeds is $\sum_{i=1}^n X_i$. The $X_i$ are not independent. However,  by the linearity of expectation,
$$E(Y)=E(X_1+X_2+\cdots+X_n)=E(X_1)+E(X_2)+\cdots +E(X_n).$$
But $Pr(X_i=1)=\dfrac{1}{i}$. This is because among the first $i$ cars, each has equal probability of being the slowest.  So the required expectation is
$$1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}.$$
