Geometric Probability at the Airport
A taxi arrives at an airport between and to wait for a particular man. It will wait for ten minutes, and if the man doesn't come, it will leave. That same man will get off his flight at anywhere from to . It will take him ten minutes to walk to were he can catch his taxi. He will wait twenty minutes for the taxi. If the taxi doesn't show up, he will rent a car and drive home. Let be the probability that the man has to rent a car where and are coprime, positive integers. Find .
The answer is 23.
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The man will get off the airplane at 2:50-3:40 and it takes him 10 minutes to walk to the taxi stand. Therefore, the man will arrive at the taxi stand somewhere from 3:00- 3:50. The chance that the man will get a taxi at 3:00 is 1/3, as the taxi can arrive at any time between 3 and 4, but the man will only get the taxi if it arrives sometime between 3:00 and 3:20.
If the man arrives between 3 and 3:10, we can see that the probability he will get the taxi increases from 1/3 and will continue to increase, as the man's arrival time gets closer to 3:10. This happens, as it is now also possible for the taxi to also arrive before the man, meaning that the probability that the man will get the taxi increases. The probability keeps on growing at a steady rate until 3:10. If the taxi arrived before the man, it will still wait 10 minutes for him, and as before, the taxi can arrive as late as twenty minutes after him. That is range of thirty minutes and the range of time that the taxi can arrive at is 1 hour. Therefore, the probability the man will get the taxi at 3:10 is 1/2.
This probability stays the same between 3:10 and 3:40, as the range of time the taxi can arrive in and still pick up the man stays at 30 minutes. (Although the man's arrival time is pushed back, the earliest possible arrival time for the taxi is also pushed back, as it can arrive no more than 10 minutes before him.)
Between 3:40 and 3:50, the probability decreases at the same steady rate it increased at between 3 and 3:10. The taxi will arrive at 4 at the latest, so the man can no longer wait for twenty minutes at the most and the range of time that the taxi can meet the man in decreases.
At 3:50, the latest possible arrival time for the man at the taxi stand, the probability he will meet the taxi cab is 1/3, as the taxi can arrive anywhere between ten minutes earlier than the man to ten minutes later. (That is a range of twenty minutes, which is one-third of the hour-long span that the taxi can arrive in.)
So, let us get our information straight. At 3:00, our man has a 1/3 chance of meeting the taxi, which increases to 1/2 by 3:10, stays constant until 3:40 and then decreases to 1/3 at 3:50. To make it easier, let us graph this. Let us graph the x-axis as hours. (0 is 3:00, 1/6 is 3:10, 5/6 is 3:50, etc.) The maximum probability that the man and taxi will meet is 1. Therefore, the area of the total probability from 0<=x<=5/6 is 5/6 * 1 = 5/6. Now, graphing our figure, (you can see a poor graph at this link ) we can calculate that its area is 7/18. The probability the man will meet the taxi is (7/18)/(5/6) = 7/15. Therefore, the probability he will not meet the taxi and must rent the car is 1- (7/15)= 8/15. 8+15=23.