# Question:A car racing challenge

## Question:A car racing challenge

Maple

Suppose N numbered cars start a race with starting positions N < ... < 4 < 3 < 2 < 1, car number 1 starting in front. (the car's index is the position on the starting grid) The race covers a total of L laps.

If two successive cars are in the position i < j, the probability that i overtakes j within 1 lap is P(i,j). This probability is independent of the position on the track and is constant throughout the race. For instance, if the starting positions are consistent with the typical car performance, we would have P(i+1,i) < P(i,i+1). These probabilities P(i,j) are constant parameters of the problem.

What is the probability that car N wins the race?

What is the probability that car i ends in position k given that car j ends in position l ?

How would you go about modeling this race? Is this a Markov Chain, a queuing model, something else?

I would imagine that it is a classical problem, is it?

thanks for any pointer, textbook, article, whatever. Illustrations for given values of N and L and the Ps are welcome, e.g. N=3, L=10, P(i,j)=p.

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