Problem: Samir and Naomi both arrive at a cafe at a uniformly chosen random time between 9am and 10am. What is the probability that they arrive within ten minutes of each other?
This is a problem of the applied probability course. The explanation used a square to visualize the probabilities, I went at it with a circle and came to 10.9/36 instead of 11/36 and was wondering if I am missing something or my approach is fundamentally flawed.
Visualizing a circle of 60 minutes with 6 equal parts of 10 minutes. Samir has a 1/60th chance of arriving at a given time, if he arrives between 9:10 and 09:50 there is a 2/6th (=10 minutes before and 10 minutes after making 2 * 1/6) chance that Naomie arrives with 10 minutes. So 40 minutes of the hour (40/60) there's a 2/6th chance that they'll meet within 10 minutes.
When Samir arrives at 9:00, there's 1/6th of a chance (10 minutes after). For the times between 9:00-9:10 and 9:50 the time they are able to meet is 10 minutes + 1 minute for every minute after 9:00 and 9:50. So there's 12 minutes they can meet at 9:02 and 9:52 with a chance of 1/60th for each minute. Since this happens at both ends of the hour it's for every 2/60th minute. So this adds up to 10/60+11/60+12/60 .... 17/60+18/60+19/60 = 145/60.
All this adds up to
P(09:10-09:50): 40/60 * 20/60 = 800/36.000 P(09:00-09:10 and 09:50-10:00): 145/60 * 2/60 = 290/36.000
800/36.000 + 290/36.000 = 1090/36.000 = 10.9/36.000
Did I miss something, did I make a mistake in my calculations or is there something fundamentally flawed about my reasoning?
I apologize for my less then concise way of explaining my method, I'm just an enthousiast and it takes skill to not have to elaborate!