A certain road has a path of a perfect circle with a single entrance/exit. A woman enters the road and walks the full circumference at a constant speed without stopping or changing direction. During her time on the road, N cars, each at its own random time during the duration of the walk, enter the circle. Each car proceeds, on the shortest path, to its own randomly selected stopping point on the circle. If cars travel 10 times as fast as the woman walks, answer the following:

1) For N=1, what is the probability that the woman “encounters” a car?

Definition: An “encounter” is when a moving car either overtakes the woman in the same direction or passes her while going in the opposite direction. If a car is stationary, there can be no encounter.

2) What is N such that there is at least a 75% chance of encountering a car?

3) For N=20, what is the expected number of encounters?

(In reply to A quibble by Steve Herman)

Indeed, you are right, and in fact, off the top of my head I think that the car must be going at least twice as fast as the woman to bring the probability down to 1/4. It would be strange if the probability were to suddenly change from 1/2 to 1/4 immediately as the car speed changed from under the woman's speed to over the woman's speed.

Posted by Charlie on 2021-02-23 21:10:29