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Circular Reasoning (Posted on 2021-02-23) Difficulty: 4 of 5
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?

No Solution Yet Submitted by Kenny M    
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Some Thoughts A quibble | Comment 2 of 3 |
Bravo, Charlie!  I agree with all of your answers.

However, i have a quibble with your comment at the end "Apparently these are independent of the ratio of the car speed and the woman's speed, as the difference of one side or the other of the circle merely accounts for an equal probability change one side or the other of 1/4, as the result above is the same as for a car of infinite speed."

That is not true, I think, if the car is slower than the woman.  In the extreme case, a car that is infinitely slow, she "encounters" every car that is going in the opposite direction, for a probability of 1/2, not 1/4.

  Posted by Steve Herman on 2021-02-23 13:53:09
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